PhD Stéphan Pezé
email: stephanpeze@yahoo.fr

Thinking the unthinkable: managerialization of work related suicides as a framing process

Keywords: Managerialization, managerialism, frame analysis, framing processes, work related suicide

Managerialism has been studied in many ways, including diffusion of management techniques notably in public sectors, management ideology or the colonization of everyday life by management thought. However, the process leading to managerialism, namely managerialization, has been largely unexplored in this literature. This article draws upon framing theory to better understand managerialization processes. The article offers two case studies that investigate following theoretical assumptions: (1) managerialization acts as a framing process on every kind of issue; (2) managerialization has framing process characteristics. Two frame analysis based on newspaper articles about work related suicide show that one of the identified frames has strong managerial characteristics and suggest that managerialization took place as a set of framing processes. Taken together, these studies provide insight to characterize the process of managerialization. Research implications and limitations are discussed in order to highlight directions for futher researchs.

Managerial discourse has permeated our society. It promotes a rational way of making things in order to achieve efficiency and performance. For some authors, this specific discourse, named managerialism, recovers an obvious rationality necessary to improve organizations as well as individuals and societies, for the benefit of the greatest number. Managerialism is not confined as a technical discourse used in business schools or organizations. As an efficiency discourse, it is used in order to help people to deal with their time, their stress and other aspect of their life (e.g. Mc Dermott and Shircore, 1999). Other authors are more suspicious and challenge this assumption. They underline problematic effects of managerial discourse's diffusion into everyday life (e.g. [START_REF] Garsten | How To Become Oneself: Discourses of Subjectivity in Postbureaucratic Organizations[END_REF][START_REF] Grey | A very short, fairly interesting and reasonably cheap book about studying organizations[END_REF]Hancock and Tyler, 2009a).

Previous works have studied managerialism as an ideology, a set of business-oriented techniques or a spreading way of representing and acting over every kind of issue and problems. If these studies help us to define what managerialism is and in what extent its effects should be critically interrogated, only few things are known about the process of managerialization. Several studies show us that management knowledge or practices diffusion is complex and not mechanistic. Indeed, some actors named senders -mainly U.S. experts of XX th century -are promoting management discourses and techniques to receivers who could adopt or adapt them. Receivers also can resist or reject them, depending on actor's ideologies, countries and historical periods. But these studies are generaly based on the diffusion of specific management knowledge and practices into organizations. They focused on already managerialized issues in a small but important part of our societies. I still don't know so far how other kind of issue, such as time or personal conduct, are being managerialized before being diffused through society in a broader sens. This paper's aim is to better explore this process in taking into account a broader meaning of management as a « distinctive series of often extremely complex ways of effecting representations and conducting interventions. » (Grey, 2009:61). To achiveve this, I propose to draw our inspiration from framing theory, and especially studies on collective action frame in social movement literature. These studies indeed look at the way social actors build and diffuse « action oriented sets of beliefs and meanings that inspire and legitimate the activities » of a movement in order to achieve specific goals (Benford and Snow 2000:614).

In this direction, I propose to consider managerialization as the mobilization of management categories (a set of beliefs and meanings) in order to frame any kind of issues or problems for the benefit of specific social actors who could act accordingly to their interests. To do so, I review framing theory in order to build those propositions: (1) managerialization acts as a framing process on every kind of issue; (2) the managerialization process has framing process characteristics.

I realize two case studies to explore these two propositions. The first case study is build around the emergence of the suicides related to work issue. This case is chosen because of its novelty and non-managerial character. By using newspaper articles, I realize a frame analysis in order to explore whether frames used to think work related suicides has managerial characteristics. I find three frames built around the work related suicide issue. Whereas two frames are built in opposition -respectively "work related suffering" and "individual's weakness" frames -the third one, namely the "psychosocial risks management" frame, has strong managerial characteristics. Indeed, it is imbued with managerial vocabulary and overall logic. The second case study pushes further my analysis. I explore the work related suicides in a single firm: France Telecom, which was a major concern of french newspapers since mid-2009. The "psychosocial risks management" frame is also identified in this case study. I analyze this managerialization process in order to highlight whether it is readable thanks to framing theoretical insights. I find strong empirical pieces of evidence that the managerialization of work related suicides at France Telecom has framing processes characteristics.

This paper contributes to the debate about managerialism and managerialization in the following ways. First, my findings show that framing theory could provide valuable insights to think managerialism processes. In both case studies presented here, a single issue is framed in three main ways. It appears that one of these frames has strong managerialism characteristics. Secondly, the results show that managerialization has framing process characteristics. I could now offer a description of managerialization as a contested process, developped through several strategies in order to counteract other frames and grow on in debates.

This paper is structured as follows. I'll first provide a review of managerialism in organizational studies and the appropriateness to cross-fertilize these works with framing theory. Secondly, I'll discuss successively my two case studies and related method, findings and discussion. I'll conclude in a final section with limitations and directions for futher researchs.

Managerialism in organizational studies

Organizational studies and scholars are referring to managerialism in many ways. I classify them in three main sets: (1) the development of new business practices and their diffusion, notably in the public sector; (2); the spreading ideology of management and (3) a critical view of managerial colonization of all kind of issues, not only in organizationl context. First of all, managerialism is used to design the promotion of managerial techniques such as TQM, business process re-engeneering, empowerment and so on. One key idea that links all these studies is that management techniques are believed superior than others, especially in the public and administrative sector [START_REF] Deem | Management as ideology: The case of "new managerialism" in higher education[END_REF][START_REF] Osborne | Reinventing Government[END_REF].

Then, managerialism could be defined by a « privileging political and economic measures of success » (Glow and Minahan, 2008:134) based on the adoption of « organizational characteristics, such as organizational forms, technologies, management instruments and values that originate from the private sector organizations » [START_REF] Smeenk | Managerialism, Organizational Commitment, and Quality of Job Performances among European University Employees[END_REF]. Besides this "business-like management" pattern, I could add two characteristics: customerorientedness and market competition ideal [START_REF] Kickert | Public governance in the Netherlands: an alternative to Anglo-American managerialism[END_REF]. Studies explicitly or implicitly built around this definition are generally dealing with managerialism from a national or broader context (e.g. studies on the New Public Management reform) or from a single organizational or sectorial level (e.g. the influence or process of managerialization). These studies show that managerialism strikes all public sectors like education and university (e.g. [START_REF] Saravanamuthu | The university in the New CorporateWorld[END_REF], health care services [START_REF] Cribb | Reconfiguring Professional Ethics: The Rise of Managerialism and Public Health in the UK National Health Service[END_REF], arts (e.g. [START_REF] Glow | I'm Here to Save You": A Case Study of the Arts and Managerialism in Australia[END_REF], NGOs (e.g. [START_REF] Roberts | NGOs and the globalization of managerialism: A research framework[END_REF], etc. Other studies follow a performance perspective and look at the efficacy of this managerialization (e.g. [START_REF] Smeenk | Managerialism, Organizational Commitment, and Quality of Job Performances among European University Employees[END_REF].

Other scholars are more skeptical and challenged the assumption that management is a universal model better than others. Indeed, this managerial lionization doesn't take into account the different types of organizations, their activities, the craft importance, their specific goals and values conflicting with a strict economic view (e.g. NGOs or public policies), the cultural traditions and so on. In short, it ignores numerous factors that called for a large range of solution to govern organizations in complex environments [START_REF] Glow | I'm Here to Save You": A Case Study of the Arts and Managerialism in Australia[END_REF][START_REF] Kickert | Public governance in the Netherlands: an alternative to Anglo-American managerialism[END_REF]. In doing so, on the contrary of its promise, the managerialization of all type of organizations leads to unappropriated governance solutions, that is softly called "managerialism contradiction" [START_REF] Smeenk | The effects of HRM practices and antecentents on organizational commitment among university employees[END_REF]. However, this conception of managerialism is strongly dependant of occupational issues and specific management practices. The differences of level and the large range of these practices fragilized the definition of what is managerialism. I will now turn on a second set of studies which allowed us to push further this conception of managerialism.

In the second set of studies, « [t]he most obvious meaning would be to understand managerialism as an ideology that secures and legitimates the interests of managers as a social group. » (Grey, 1996:601). This analysis is well develloped in Anthony's works (1977). By this view, managerialism is a set of control tools to the benefits of managers.

Managerialism is disguised in scientific language which produces unity of purpose and values. This allows the integration of organizational membres and maintains the empowerment of managers over them. But this conception raises critics because several key terms of this concept are subject to intense debates. For example, are managers a single social group, an elite or a class with international and transcultural similarities, concerned with taking power over employees? This critic is overcomed if I move from who manages to social construction of manageability (Grey, 1996:602). To that extent, it is well accepted that managerialism is « the generalized ideology of management » (Parker, 2002:10). This turns us to another set of studies.

Finally, managerialism could be seen « in terms of the social consequences of a particular way of looking at the world which stresses, first, that the world is manageable and second that the world should be managed. » (Grey, 1996:601-602). More precisely, it implies « the almost direct transference of the imperatives, logic and values associated with managerial expertise […] into cultural ressources » associated with every aspect of our occupational and mundane life (Hancock, 2009:11-12). This analysis takes place in a long range of studies following Weber's initial works about world rationalization (1968). The operator of this rationalization, management, is qualified as a dominant expertise that pervades into society, producing oppressive effects. Hidden under apparent neutrality, management is a form of control over labor and social relationships. Authors challenge this lionization of management as the solution to all economic, social and self-oriented problems. Management is becoming a problem [START_REF] Parker | Againts Management: Organization in the age of managerialism[END_REF]. In this approach, management is clearly criticized for its pretentiousness to be the only « one [response] that is directed at resolving the implacable contradictions of what is, in effect, an irrational mode of subjective existence (mass mediated, time pressured, performative and so on). » (Hancock and Tyler, 2004:631). This approach goes further than previous ones because it doesn't address only managerial techniques or their diffusion in specific public sector, nor that it seeks to find who the "master of puppets" is. It embraces management itself and its colonization of every kind of topics.

Critics of managerialism are focusing more on management's diffusion as the only way to do things than on management itself (e.g. [START_REF] Costea | Managerialism, the therapeutic habitus and the self in contemporary organizing[END_REF][START_REF] Deetz | Democracy in an Age of Corporate Colonization[END_REF][START_REF] Grey | Toward a critique of managerialism: the contribution of Simone Weil[END_REF]2009;[START_REF] Hancock | Management and colonization in everyday life[END_REF][START_REF] Parker | Againts Management: Organization in the age of managerialism[END_REF]. Everything, every human and natural domain seems to fall under the realm of management. Critics underline the effects of this process. Under a performative imperative, reduced to a calculable individual, people follow a strict rationality that strongly reduces the possibilities of human existence [START_REF] Grey | A very short, fairly interesting and reasonably cheap book about studying organizations[END_REF]. This managerialism doesn't concern just the workplace [START_REF] Hancock | MOT your life': Critical management studies and the management of everyday life[END_REF]. For example, there is a managerialization of everyday life which concerns sexuality, health, sleep, domestic issues, retirement and so forth through managerial gurus, "how to" books, self-help coachs or lifestyle magazines [START_REF] Cullen | How to sell your soul and still get into Heaven: Steven Covey's epiphanyinducing technology of effective selfhood[END_REF]Hancock and Tyler, 2009a). Futhermore, managerialization embraces the self and subjectivity. Our inner thoughts and actions become subject to managerialism, « leaving little room for an 'authentic', irrational or simply 'unmanaged' experience of everyday life in which we have the opportunity to feel that we can just be, or rather become, ourselves. » (Hancock and Tyler, 2004:640). In and out our day-today working life, managerialism produces a normative set of principles that requires selfwork to improve ourselves, relying on the "therapeutic habitus" of our time [START_REF] Costea | Managerialism, the therapeutic habitus and the self in contemporary organizing[END_REF]. This knowledge of the self is instrumentally used to enhance individual performance -and finally, it is hoped, the organizational one -leading to the production of a type of conduct close to a "corporates athletic" style [START_REF] Kelly | New Work Ethics? The Corporate Athlete's Back End Index and Organizational Performance[END_REF]. That is to say that managerialism is a way of increasing control for private and limited interests, notably corporations, whatever inequalities result. However -and fortunately -managerialization is not a totalizing process, « such managerial incursions into everyday life are uncertain, contradictory and sometimes even incoherent » (Hancock and Tyler, 2009b:xii). Resistance and subversion are still possible inside (e.g. [START_REF] Fleming | Contesting the Corporation: Struggle, Power and Resistance in Organizations[END_REF] and outside the workplace (e.g. [START_REF] Thanem | The New Public Health and the Struggle to Manage Healthy Bodies[END_REF].

To sum up, managerialism is a key word used to speak about the process of the diffusion of management categories in order to think about and control all aspect of life and world and to denounce oppressive effects resulting from this process. However, we know little about managerialization as a process. Few elements could be found in previous studies. Some case studies of the managerialization of single firms over time tell us that this process is a complex dynamic, depending on elite actors and their interactions, both inside and outside organizations (e.g. State institutions), their ideological and professional underpinnings, the national business system, etc. (McLean et al, 2007;[START_REF] Mueller | We are all managers now': Managerialism and professional engineering in UK electricity utilities[END_REF]. Managers, consultants and academics are also implicated in this process, trapped in « an unholy (but well-compensated) trinity of self-interest and back-slaping » (Parker, 2002:2). Management has also been studied at a major scale as the growing power, during the XX th century, of new professional experts coming originally from U.S. and their models and discourses in shaping social relations and action [START_REF] Frenkel | The Americanization of the Antimanagerialist Alternative in Israel[END_REF]. Their results are valuable. Indeed, these studies identify how senders, receivers and carrying organizations are interacting in the diffusion of management knowledge or practices. The diffusion process is not mechanistic. Receiversindividuals, groups or organizations -could adopt, adapt or resist and reject them. Moreover, these reactions depend on specific cultural context such as the role of State or elites ideologies in this diffusion and reception process [START_REF] Kipping | The transfer of management knowledge to peripheral countries[END_REF]McLean et al., 2007;[START_REF] Üsdiken | Exporting Managerial Knowledge to the Outpost: Penetration of 'Human Relations' into Turkish Academia, 1950-1965[END_REF]. But these approachs are generaly focused only on the diffusion of existing knowledge processes or particular managerial practices. Moreover, they concern specific contexts such as organisational or scholarly actor's reactions to these diffusion processes. Up to now, we are not able to embrace a broader view of managerialization, i.e. how an issue is previously being thought with managerial patterns, and how this managerialized issue is diffused into broader society. What is in play here is twofold: avoiding the totalizing temptation of seeing managerialism everywhere and also exploring the protean caracter of managerialism, trying to catch under the heterogeneity of meanings and levels of analysiseven its contradictions -of some common patterns. In order to contribute to answer this gap, I propose to look at alternative bodies of work. Indeed, it could help us to draw general characteristics of managerialism viewed as a process.

According to Grey, « management consists of a distinctive series of often extremely complex ways of effecting representations and conducting interventions. » (2009:61). So managerialization provides us some ways of knowing the world, but also connected specifics manners of influencing it. This assumption gives us a key characteristic about managerialization: ways of thinking. It is also close to the concept of framing that have been extensively studied as a process. Indeed, studies on framing look at the various ways used by social actors to construct and diffuse « action oriented sets of beliefs and meanings that inspire and legitimate the activities » of a movement in order to achieve specific goals (Benford and Snow 2000:614). Such a frame only exists in the relationship with a specific issue or event around which framing processes take place [START_REF] Entman | Projects of Power: Framing News, Public Opinion[END_REF]. Due to this nearness, could framing studies provide us valuable insights to think about managerialization? For the purpose of exploring this possibility, I will now present the framing literature. Then I will draw upon it in order to think about managerialization in a cross-pollination perspective.

When framing theory meets managerialization…

First of all, I will present the framing literature before bridging it with managerialism.

Basics of framing theory

The concept of framing is heterogeneous. It is used in several fields (management, sociology, cognitive psychology, political science, etc.) and is referring to multiple analysis methods and objects (linguistic and discourse analysis, social movements, media impact, etc.). In this section, I focus on cognitive frames, that is « knowledge structure (frames as cognitive representations) » (Dewulf et al., 2009:156) and especially cognitive issue frames, i.e.

« heuristics that people use to interpret a situation » (Dewulf et al., 2009:167). Individuals give sense to their day-to-day reality through thoses frames. Rituals, conventions, speeches, etc. are all social experiments in which frames are created and shared to give direction and guide action (Creed et al., 2002a). Following Goffman, thoses frames are "schemata of interpretation" (1974).

Collective action frames. One of the main approaches of framing is "collective action frames" where scholars seek to identify the shared meaning negotiated collectively, mainly by social movements, in order to trigger action [START_REF] Benford | Framing processes and social movements: an overview and assessment[END_REF]. Their function is to mobilize the potential of adherents and to ensure homogeneous collective actions, but also to recruit new adherents and to counteract opponents. These frames are built from diverse elements, ideas and symbols, assembled in a coherent way through a discursive process.

However, this organization is not clearly visible. This is mostly composed of underlying principles took for granted by movement adherents. The ultimate aim of the diffusion of these collective action frames is to influence other people. According to [START_REF] Hunt | Identity fields: Framing processes and the social constmction of movement identities[END_REF], there are three categories of people in front of a frame: (1) the protagonists or proponents, which find the frame appealing; (2) the antagonists or opponents, opposed to the frame and its related actions; and (3) an audience of hesitant people which could possibly become protagonists or antagonists. What is at stake here is this audience. Several framing processes set up in order to shape values and beliefs of this audience have been identified.

Framing processes. Numerous works are dealing with framing processes. It is difficult to classify and structure all the processes identified because they differ in level, scope or goals.

In order to present them, I will follow Benford and Snow's review (2000): first I will have a look at the generation and development processes in order to answer the question: where are frames coming from and how are they growing? Secondly, I will have a deeper exam of diffusion processes in order to understand how frames are getting in touch with people and are spreading into society.

In their review of framing processes and dynamics, [START_REF] Benford | Framing processes and social movements: an overview and assessment[END_REF] have identified a first set of processes concerning frame initiation and developpment: (1) discursive processes; (2) strategic processes or frame alignment processes; and (3) contested processes.

First, « [d]iscursive processes refer to the talk and conversations -the speech acts -and

written communications of movement members that occur primarily in the context of, or in relation to, movement activities. » (Benford and Snow, 2000:623) Basically, at this level, components of the frame are selected and assembled. Snow and Benford found very little research on these processes.

Secondly, many empirical works have been done regarding a second range of processes labelled as "strategic", i.e. « deliberative, utilitarian, and goal directed » (Benford and Snow, 2000:624). [START_REF] Snow | Frame alignment processes, micromobilzation, and movement participation[END_REF] have indentified four basic alignment processes in these strategic attempts: frame bridging; frame amplification; frame extension; and frame transformation (see table no 1 for main characteristics).

Frame alignment processes

Main characteristics

-Frame bridging

Linking two or more structured frames around a particular issue; frames need to be reconciliable Benefits: recruitment of new members which are ideologically close to the group but not primarily concerned about the issue targeted; allows a group to get in connection with others that are not primarily concerned about the issue targeted but are "frame reconciliable" (e.g. [START_REF] Gerhards | Mesomobilization:organizing and framing in two protest campaigns in West Germany[END_REF]McMallion and Maines, 1999) Considered as the most prevalent framing strategy [START_REF] Benford | Framing processes and social movements: an overview and assessment[END_REF] -Frame amplification Highlighting values and beliefs of the frame which are found in dominant culture's values and beliefs Benefits: reach a more justifiable and credible position when values and beliefs underlying the frame are consistent with common wisdom (e.g. [START_REF] Berbrier | Half the battle': cultural resonance, framing processes, and ethnic affectations in contemporary white separatists rhetoric[END_REF] -Frame extension Extending the range of issues or problems targeted in a group or social movement Benefits: recruitment of new members sensitive to new issues targeted Several empirical studies suggest that this extension is the product of a discursive process that could potentially fragilize the homogeneity and solidity of groups or social movements concerned [START_REF] Benford | Framing processes and social movements: an overview and assessment[END_REF] -Frame transformation Changing radically the underlying meanings of the frame in order to replace them by new ones more attractive Benefits: reach a more attractive position; move from accepted beliefs to new image of a movement or an issue (e.g. [START_REF] White | Talking black: micromobilization processes in collective protest against rape[END_REF] Thirdly, beyond these strategies brought to achieve goals of the frame and its proponents, a large body of empirical work shows that frames are the product of contested processes. It means that the relative homogeneity of a frame is never imposed by leaders of a group or a social movement. The frame is a collective construction achieved by contestation. The contestation could occur because of counterframing attacks from frame opponents [START_REF] Benford | Framing activity, meaning, and social movement participation: the nuclear disarmament movement[END_REF] but also because of internal disputes into the frame [START_REF] Benford | Frame disputes within the nuclear disarmament movement[END_REF]. Finally, this contested process is also mentionned to qualify the abductive fit that occurs between a frame and its product, i.e. collective action (e.g. [START_REF] Ellingson | Understanding the dialectic of discourse and collective action: public debate and rioting in antebellum Cincinnati[END_REF].

A second set of processes indentified by [START_REF] Benford | Framing processes and social movements: an overview and assessment[END_REF] concerns frame diffusion.

To date, few studies have explored this topic. The diffusion of a frame refers to the dissemination of it in other movements or cultural environments, which raises a question of compatibility. Two ideal types of diffusion could be distinguished [START_REF] Snow | Alternative types of cross-national diffusion in the social movement arena[END_REF]: strategic selection (an active adopter intentionally selects a frame and imports it into his/her own cultural environment) and strategic fitting (an active adopter will intentionally selects a frame and adapts it into his/her own cultural environment). In both cases the diffusion processes described imply active agents and a strategic intent. There is another kind of diffusion, more passive, through which people get into connection with frames through mass media. This kind of diffusion is more addressed in studies about the effects of framing on public opinion.

Framing effects. The diffusion of frames trough mass media has an impact on the public opinion. This "framing effect" is « the process by which people develop a particular conceptualization of an issue or reorient their thinking about an issue. » (Chong and Druckman, 2007:104). First of all a frame only exists in relation to a specific issue or event [START_REF] Entman | Projects of Power: Framing News, Public Opinion[END_REF]. Once it is diffused through mass media, its influence is dependent on multiple variables: strength, relevance and repetition of a frame, but also its cultural compatibility (i.e. a frame is easier accepted if it respects general moral principles). The strongest and most pervasive the frame is, higher the framing effects are.

It also depends on the other available frames. A frame is indeed never alone, several frames are competing around a single issue [START_REF] Sniderman | The structure of political argument and the logic of issue framing[END_REF]. However « little is known about the dynamics of framing in competitive contexts » (Chong and Druckman, 2007:113). It seems that individuals are sensitive to frames that are congruent to their deep values [START_REF] Benford | Framing processes and social movements: an overview and assessment[END_REF] but they also can be convinced by opposed frames if their argumentation is more consistent, i.e. by the strength of the frame [START_REF] Chong | Framing theory[END_REF]. only a private and isolated one (1897), work related suicide is a new social issue for most westerner countries. Initially those events were confined to few occupations, as farmers facing debts, prison guards or policemen. However, since the end of the 90s, the range of hit professions is growing [START_REF] Bègue | Suicide et travail : que faire ?[END_REF]. The novelty of this issue is an advantage for this study: thanks to its relative virginity, few people have ever thought about it. It opens the door for a frame analysis which is not contaminated by a specific historical or ideological ground. For instance this issue has not really been exposed to managerialism.

Moreover it is an extreme analytical lens. This extreme choice is relevant for my study because these events are concrete. Social actors are facing a precise problem and have to say something about it. By doing so, we avoid some difficulties linked to the polysemia of other wider issues. Furthermore, as research object, the extreme character of these events allows us to see easily frames and actors behind them. It acts as a trigger. The death of a person, especially when (s)he commited suicide, raises indeed lots of questions. There is a debate about the form's responsibility and the influence of working conditions in this act. On this kind of issue, people will produce clear discourses, easier to identify -as well as actors which produce these discourses. Finally, facing this kind of violent and irrational event, people are looking for answers and sensemaking. Discourses about these issues are highly emotional and symbolic. There are generally highly controlled before being diffused to the public opinion. It raises its interest for the study of framing strategies. Before detailing my data analysis and collection method, here is an overview of this case which takes place in France.

Case overview: the work related suicide genesis (2006)(2007)(2008). In February 2007, several newspapers have reported three suicides commited in the same workplace since two year: the Renault' research site. Between October 20, 2006 andMarch 13, 2008, five suicides hit this automobile manufacturer. They are extensively reported in media. Missing persons have commited suicide in the workplace, for example by jumping from a building, or at home -with a letter underlying work difficulties. Some trade unions of the firm have blamed bad working conditions and asked their top managers to react. In the same time, 600 to 800 employees have demonstrated in memory of their co-workers. The CEO of the compagny, Carlos Ghosn, has developped actions at the national level in order to help the distressed persons. 350 persons were hired, notably 95 "closeness managers".

On July 16, 2007, 55 years old worker hang himself in the factory of another french automobile manufacturer: PSA Peugeot-Citroën. It was the 6 th suicides that occur in this firm since the beginning of the year. The compagny has decided to realize stress audits in 3 of its plants in order to better understand and assess stress factors, and to set up a psychological helpline. The CEO of the compagny, Christian Streiff, rebelled against the amalgam which is made to say that work kills.

Several cases have been reported during the same period in other firms: EDF, La Poste, AREVA, Sodexho, Ed, IBM France, France Telecom, HSCB, BNP Paribas, etc. No business is immune. These events had a great echos in public opinion. Lots of reportage had been undertaken in newspapers and television. This has a great bulk of effects: french national insurance system has classified several suicides as occupational injuries; futhermore, the french Minister of Labor has told himself concerned about these event. He asked national trade unions to discuss the translation of a European agreement on workplace stress. They did it on July 2, 2008. All those events had created a special sensitivity about occupational mental health issues, especialy on workplace suicide, but more generaly on stress, harassment and quality of workplace life.

Data collection

I have used newspaper articles as data. Frame analyses are commonly using these data sources [START_REF] Chong | Framing theory[END_REF]. The macro level of analysis is relevant to observe the constitution of a plurality of frames. Indeed, through mass media, lots of different kind of people and opinions are provided.

I have collected articles from four main french newspapers: Le Monde, Les Echos, Libération and Le Figaro. They cover a large range of political opinion, from left to right wing. By using the Dow Jones Factiva database, I have looked for articles published between February 1 st , 2007 (first mention of work related suicide) and April 31, 2008. That last date was stopped from this point because of a good semantic saturation. I used key words (suicide; workplace suicide; work related suicide 1 ) and have read all the articles in a systematic way in order to select relevant articles. Some articles were excluded because they were dealing with other topics (e.g. articles on suicide bombing attacks). The frame analysis detailed below was conducted on the base of 71 newspaper articles 2 .

Data analysis

I have conducted a classical frame analysis following Creed et al. principles for organizational research (2002a;2002b). According to them, frame analysis « is a technique for approaching a text by attending to its diverse idea elements with the following question:

What holds these elements together? » (Creed et al., 2002a:37). I have used their "signature matrix" based on previous works of framing scholars (e.g. Gamson and Lasch, Snow and Benford, etc.). This signature matrix, presented in table no 2, « enables analysts to enables the analyst to (1) sort the idea elements so that connections among them can be discerned,

(2) identify and distinguish among the different unifying structures or frames that hold them together, and (3) discover how diverse idea elements are deployed in integrated ways. » (Creed et al., 2002b:482). Each categories of the matrix were completed in an abducive way. It is noteworthy that newspaper articles are made of elements coming from several potential frames. My first task was to seek for adapted patterns to separate them. So I began by looking at few matrix categories -problem definitions and problem elaborations -in order to produce a first set of coherent discursive elements. These potential frames are built thanks to a first complete reading of the documents collected. Having done this, I continued the analysis by exploring the other categories for each temporary frame through a second complete reading. This operation led us to regroup some frames into one because of their proximity. At the end, I kept only three significant frames and named them according to their underlying logic. These three frames are detailed below in order to answer my research questions.

Findings

I found three frames in newspaper articles. I named them according to their main characteristics: "work related suffering", "individual's weakness" and "psychosocial risks management" (see table no 3). this question, I will consider successively whether this frame is infused with (1) managerial markers and (2) managerial logic.

First of all, we must wonder if this frame presents markers of managerial categories (i.e.

performance imperatives, the will to control, instrumental rationality, managerial buzzwords, etc.). Conversely to the others, this frame is full of managerial markers For example, "policy"

or "plan of action", "inquiries", "assessment" or "survey" are repeated several times;

"management" itself is profusely used, as well as "recruitment", "human ressource management", "observatory", "concrete measures", "sustainable devices", setting up "new actions" or "new measures", "deep reflection", "meetings", etc. This is not surprising because of the status of proponents -people who are interviewed into newspapers: mainly top managers and consultants. However, this is not only words. Indeed, a real effect is to frame the problem causes and solutions into managerial terms. For example, causes found by organisms in charge of inquiries are linked to specific managerial tasks and not to overall news modes of management (as in the frame named "work related suffering") or to individual weakness (as in the second frame This managerial logic could be resumed as:

a fundamentaly ideological discourse […] that directs individual decision-making and socially embedded practices towards an overriding prioritization of instrumentally orientated action, combined with an almost heroic valorization of individual entrepreneurialism, both of which are presented as a universalistic panacea in the face of the uncertainties of everyday life. Moreover, it is one that frequently evokes the language and symbolism of an idealized version of management practiceparticularly the values of quantification, strategic planning and the marketing function -combined with a justificatory appeal to the efficiencies of the free market economy. (Hancock, 2009:7).

This general logic is found in the frame. Indeed, in these explicit quotes, we could see a will The aim of this second case study is to identify if patterns of managerialization are better understood with frame's development and diffusion theory. To do so, I have identified (1) whether or not a managerial frame also appear in this case study and ( 2) to what extant this frame is thinkable in framing theory categories. I first have proceeded to a frame analysis just as I did in the first case study. I found a managerial frame which has the same characteristics thanks to a major strategic process: a successful frame bridging acheived with the stress management frame. Thanks to a frame amplification strategy armed by the constitution of a media plan, proponents profusely intervene in mass media in order to diffuse their message:

the work related suicide issue is under control by our managerial approach. My findings show that the managerialization process could be studied as the result of mixed sub-processes that take place simultaneously and influence each other.

Conclusions

I have realised two case studies which show that managerialization acts as a framing process on the work related suicide issue and that this real process of managerialization has comparative framing process characteristics. Diverse elements, ideas and symbols picked from managerial though, are assembled in a coherent way and shape a frame named "psychosocial risks management". The aim of this frame is to act upon work related suicides in order to solve it. Proponents of this frame seek to give it some strength in order to acheive their goals. To do so, they implement several strategies of frame alignment. These strategies are however mostly emergent and dependant on discursive and contested processes. Indeed, internal debates, counterframing attacks from opponents and other criteria play a key role in the shaping of managerialization. Throughout these melting sub-processes, the frame is disseminated to an audience by proponents -mainly through mass media -and some adopters could also paly a role in this diffusion. Given the strong similarities of these characteristics, I can infer they are relevant for the managerialization of all kind of issue.

My study is not free from limitations. As I have mentionned, managerialism is quite consensuel and took for granted in our time. It could be interesting to pursue new media based analysis around several issues accross the 20st century. Some limits aslo come from the analysed data. Indeed, I have only focused on newspaper articles which provide a single (or

  to take control over the issue thanks to classical managerial style: « The group will begin today a survey on work related stress with 3,000 employees[…]. The survey will close on December 20 and its findings will be made at the end of the first quarter of 2008 to determine a "plan of action". »(Les Echos, November 19, 2007); « In addition, PSA has appealed to the specialized consulting firm Stimulus, which is leading a study whose findings will be by trade unions as the responsible of the reorganization, decided to leave the group. He was replaced by Stéphane Richard who became the new CEO six months later.Concerning legal implications, the french national insurance system has classified several suicides as occupational injuries. The french labour inspectorate administration (Inspection du Travail) made a report mentionning that some management practices had produced harassment situations and endangerment of employees. Finally, for the first time in this situation, the french court has decided to file a lawsuit for "unvoluntary homicide" against France Telecom on March 15, 2010. This second case is hugest than the first one: in the number of deceased, the political and legal implication and the internal reorganization of the top management board. It allows us the possibility to look at the workplace suicides issue from another point of view -a case study of the frame built by a single firm. It could be very helpfull to look at this case in order to highlight the progressive birth and development of the framing strategies used in order to preserve the firm reputation.Data collection and analysisI have collected articles from same french newspapers as for the first analysis. I have looked for articles published between July 1 st , 2009 (first mention of workplace suicides in France Telecom) to march 31, 2010. The frame analysis detailed below was conducted on the base of 105 newspaper articles.

Table 1 : Main characteristics of strategic framing processes (adapted from Benford and Snow, 2000; Snow et al., 1986)

 1

signature matrix (from Creed et al., 2002b:483)

	Signature matrix categories
	1. Examples of idea elements: symbols, images, Illustrations, catchphrases, etc.
	2. Problem definition
	3. Problem elaboration: diagnosis, prognosis, and motivation
	4. Inferred master frame(s)
	5. Cultural building blocks: schemas. systems of meaning
	6. Proponents
	7. Opponents
	8. Audience targeted
	9. Frame amplification
	Table no 2:

Table n°3 : framing the workplace suicides issue

 n°3

  workload, 311 were carried out to date. Social workers were recruited, a specific HRM was dedicated at the Technocentre in June 2007, and a "stress observatory" introduced. [...] The workday opening hours have also been restricted to avoid overruns »

	(Les Echos, march 13, 2008); « EDF's top management has announced several measures:
	creation of a quality of work life observatory, setting up a hotline for employees, and simplify
	administrative procedures to lighten the workload. » (Les Echos, march 18, 2008). However,
	the intensive attendance of managerial words is not sufficient in order to prove the
	managerialist nature of the frame. I have now to find whether this frame is infused with a real
	managerial logic of control over work related suicides.
	Among the main causes of anxiety, the 3,200 employees surveyed by Stimulus highlight "the
	rigid procedures and closely monitored", "lack of time", "the need to constantly adapt" or
	"lack of recognition" of their work. » (Les Echos, march 18, 2008). Accordingly, solutions
	are found in managerial adjustments: recruitments, HRM developments, etc. But I don't find

). Results of two different surveys show it well: « The firm Technologia, author of the report, said that the situations of tension are "strongly linked to the lack of recognition". This was partly due to "a lack of support in difficult situations", but also "to limited promotion prospects". » (Le Monde, October 21, 2007); « This study demonstrates that numerous stressors resulting primarily from the work organization.

here the managerial adjustments called for in other frames. For example: « At Renault, "a team's support plan" has been established. 350 new hires have been promised in order to divide the

Drawing upon framing theory to think about managerialization as a process

If managerialization acts as a frame, the process would have the following characteristics: around anykind of issue or problem, a specific way of thinking -its causes, solutions, etc.appears following a discursive and contested process. This is the creation of the frame.

Different elements, ideas and symbols picked from managerial thoughts, are assembled in a coherent way and shape a frame. The aim is to act on the identified issue or problem to solve it. This process is not blind or made by chance. There are people for and against it. Once the frame is built, proponents seek to give it some strength in order to achieve their goals. That is to say that several kinds of strategy of frame alignment could be implemented (frame bridging, frame amplification, frame extension and/or frame transformation). Then the frame is disseminated to an audience by adopters who select, import and/or adapt the frame to their cultural environment or through mass media. In this last case, proponents would try to enhance the frame's strength, repetition and overall cultural compatibility.

In order to explore whether these propositions are valids, I propose to build two case studies.

The first one will help us to answer an important question: is managerialization really readable as a framing process? That is indeed a valuable insight if we want to continue to compare the managerialization process characteristics with framing theory. The second one will help us to make this comparison. I will now detail successively these two case studies.

Thinking the unthinkable: framing the work related to suicide issue

Case overview

The choice of an extreme issue as a telltale analytical standpoint. I choose a single issue in order to underline how social groups produce frames: the "work related to suicide" issue. If suicide is studied since Durkheim's original book which shows that it is a social issue and not

The first frame identified is the "work related suffering" one. It sustain the idea that a workplace suicide is showing working conditions. It is a message send by the dead person.

« Experts are unanimous: to commit suicide on one's workplace is not an insignificant choice and is of special significance » 3 (a jounalist, Le Monde, February 2, 2007). There is no automatic melting between suicide and work but a strong interrogation around work influence. As an expert puts it: « Suicides are the end of a process of disintegration of the social framework that shapes the occupational world. A work organization may not be reducible to a division and an allocation of tasks, cold and rational, assessable at all times. » (Christophe Dejours, Le Monde July 22, 2007). The disappearance of collective dimension of work is seen as the result of work relations individualization and the development of competition between employees (changes set up in order to increase productivity). Solutions proposed are recreating social links and legal constaints upon firms so that they increase efforts toward better working conditions.

The second frame identified, "individual's weakness", is the reversal of the first one. Indeed, suicides are thought as complex individual acts of weak persons. It is a victim blaming frame.

The underlying idea is that bad working conditions do not kill. Even if these conditions have an influence, it is too difficult to categorically establish it. Moreover, these persons are generally facing strong private difficulties (divorce, death of a close, etc.). As a top manager puts it: « the link between suicide and working conditions in the nuclear power plant can not be categorically established. Every suicide is a personnal choice. [Our firm] is not able to tell what elements can trigger such a gesture. » (Libération, March 6, 2007). If the major cause is individual weakness, the solution is to detect those employees thanks to physicians and psychological helplines. Apart from this measure, there is no real solution inside the workplace. Indeed, « If we had global economic conditions more favorable, I also think that it would derease the work pressure. […] If employees commit suicide, it is mostly the fault of "global economic conditions" » (Laurence Parisot, President of employer's trade union, Le Monde, March 22, 2008).

Whereas these two frames are built against each other, the last frame identified called "psychosocial risks management" is more consensual. Working conditions could have an influence on mental health through psychosocial risks. Workplace suicide could be one of the extreme consequences of these risks -scientific studies sustain this proposition -or be totally extraneous. No one is shown as responsible: « Everyone has a responsibility: it is neither to deny nor to exploit the link with the work » [START_REF] Anact | Prévention du suicide en relation avec le travail[END_REF]. Solutions aim to avoid new dramas. They are always presented as the fruit of a collective multidisciplinary reflection, notably with trade unions: (1) just after such an event, support coworkers and prevent contagion effects (psychological helpline and strong internal communication); (2) in a preventive way, acheiving a collective approach of psychosocial risks through risk assessment and policy definition and implementation; (3) in a continuous way, follow warning indicators. Proponents are generally top managers as human ressource managers, consultants and health and safety experts (for France: INRS and ANACT, similar to UK's HSE or US's NIOSH). It is difficult to identify opponents because all previous actors are integrated in this frame into the multidisciplinary move. The frame amplification strategy used here is neutrality and the will to engage all actors around a common goal: avoid further tragedies. Skirting the issue of responsability and basing its claim on scientific studies and official agencies, this frame aims to bring together those who wish to act. This is a pragmatic position: the work entails risks that must be removed or reduced collectively in a consensual manner.

Is managerialization thinkable as a framing process? Having said this, I can now go back to my research question: is managerialization really readable as a framing process? To answer delivered at the end of the first quarter. Focusing on a representative sample of 10% of the three sites in Mulhouse,Sochaux and Velizy (30,000 employees in 130,000 in France), it aims to identify stressors, to evaluate their level, to identify categories of employees most at risk, then, to determine areas of reflection and action. » (Le Figaro, April 7, 2008). In those two quotes, the approach is the same: (1) a quantification thanks to risk assessment upon a representative sample of employees, generally entrusted to expert consulting firms; then (2) definition and implementation of an adequate policy in order to master stress factors; (3) the overall approach requires a strict strategic planning.

However, this frame is not completely managerial. Indeed, I do not find performance imperatives in newspaper articles. Futhermore, I do not find references to the market economy. This is most likely due to the issue: an internal issue (limited to organizational context that exclude competition references) infused with a high tragical burden (performance references could be quite odd there).

To conclude, several pieces of evidence show us that this frame is a managerial one.

Managerial markers and logic are deeply underlying top managers and consultant discourses about workplace suicides. However, this frame seems to be still in his elaboration stage.

Remind us that work related suicides is a new issue, infused tragically and that discourses were produced under media pressure.The incomplete character of managerialization is probably mostly due to this specific context. Now that I have established that managerialization could be seen as a framing process, I am able to go further and try to identify this process characteristics. This is the aim of a second case study.

The France Telecom affair: a multiframe processual analysis

Case overview

A single firm in the middle of the storm (2009)(2010) Under pressure, the CEO announced several measures: the recruitement of 10% occupational physicians more, the recruitement of 100 "closeness human ressources managers" and a huge inquiry about the uneasiness of employees. The inquiry was undertaken by Technologia, a consulting firm choosen by trade unions. 80 088 questionnaires have been answered by the 102 000 employees of the firm. The CEO also decided to temporarly stop a reorganization program called "time to move". The firm was engaged in a vast restructuration on the french soil: since 2005, 22 000 persons have left the firm and 40 sites have been closed. These changes had created a pressure on employees who had to move to another workplace and, sometime, their job and occupation inside the compagny. The "time to move" program, key action of this reorganization, was the target of several critics from trade unions or analysts in newspapers' columns. On October 5, the executive manager Pierre-Louis Wenes, considered as the one identified above. Then I have questionned this frame with framing process characteristics that I identified in literature: initiation through (1) a discursive and (2) a contested process during which elements are picked up and assembled in a coherent way; (3) development through oriented and volontary strategies of frame alignment; (4) diffusion of the frame through mass media and adopters who select and adapt it to their own cultural environment. I have highlighted managerial frame's elements for each of these factors. This led us to conclude about the relevance of this comparison.

Findings

First of all, in this second case study, I found a managerial frame similar to the previous one.

We can consider it as a managerial frame used to lead people to think about workplace suicides in -and to act according to -a managerial way. This managerial frame does not exist on its own. It is the result of a process, or at least of several sub-processes mixture. As my findings show, this process could be studied as follow.

Frame initiation. Once the work related suicide issue appear, managerialization begins. If we want to try to identify its birth thanks to frame theory, we must find piece of evidence of a discursive process. Framing scholars found that discursive processes come from two subprocesses: (1) frame articulation and (2) frame amplification.

(1) I found three main ways to highlight analyse the frame articulation process. First of all, some elements appear and are duplicate over time. For example, the denunciation of an instrumentalization of workplace suicides by successive Executive Managers. The first attempt in made by Mr. Wenes in September, 2009, at the beginning of the affair: « "Some have made dead people talk", accused number 2 in order to relativize the suicides. » (Libération, septembre 29, 2009). Then, this claim is consolidated by his successor Mr.

Richard in February, 2010: « I am shocked by how some instrumentalize the suicides to conduct a trial against the company. We recognized the depth of the malaise, we are definitely not in denial. The suicides continue, but take this "indicator" to try to prolong the controversy, it's indecent. » (Le Monde, February 26, 2010). Secondly, some elements appear but evolve over time. It is the case of the solutions to the suicide issue undertaken by France Through time, under political and internal trade unions pressure, the CEO decided to stop a part of the reorganization: « the directorship […] will extend the freeze of the mobility process untill the end of the year. » (Le Figaro, October 6, 2009). This measure will be followed by others which are strongly repeated over time: recruitment of occupational physicians and HR managers (September, 2009, to February, 2010), a stress agreement negociation undertaken with trade unions (September, 2009, to February, 2010), etc. As we saw in the previous case study, these elements are a strong characteristic of the managerial frame. Thirdly, some elements appear but are not kept. There is a progressive selection or refinement. For example, I found statements which claim a relativization of the importance of the workplace suicide. This relativization is based on statistics and occurs only between September and October, 2009. This fleeting claim is that there are fewer suicides in the compagny than in general population: « If we look statistics with a cool head, there are no more suicides in France Telecom than in the general population. » (Le Figaro, octobre 3, 2009).

(2) The frame amplification is mainly acheived through repetition of selected elements. I mentioned above the repetition of solutions as recruitment, etc. Between October, 2009, and February, 2010, more than 10 articles have promoted and reminded extensively all the measures implemented by France Telecom. This creates a mediatic saturation.

Frame development through contested process. Framing scholars found frame evolution is the result of a contested process. It is triggered by (1) internal disputes into the frame, (2) counterframing attacks from frame opponents and (3) an abductive fit between a frame and the collective actions it produces.

(1) First of all, the contested process is visible through internal disputes. Indeed, several newspaper articles relate that frame proponents, i.e. government and ministers, are debating with France Telecom's CEO. For example, the Minister of Economics had convened France Telecom CEO on october 1 st , 2009, in order to ask him « explanations » and « to know how he will give back reliance to employees and take problems by the horns » (Les Echos, october 1 st , 2009). Furthermore, the Minister of Labor has also convened the CEO because he was « strongly preoccupied » (Le Figaro, September 13, 2009). They met three times in the beginning of September. These interventions have contributed to push France Telecom's executive committee to move to a more managerial frame.

(2) The contested process is also visible through defensive claims which face counterframing attacks. I mentioned above some statements of frame proponents which aim to denounce or relativize blames made against France Telecom, but these elements are not really strongly part of the frame. Other counterframing attacks have more influence. For example, several articles relay opponent's statements blaming the CEO's attitude. Some left wing party representatives underline the communicative clumsiness of D. Lombard (e.g. Les Echos, September 30, 2009). Indeed, he talks about suicide as a "fashion" on September 15 and apologizes about it the day after. This had a strong influence on the decision to follow a media plan made in the beginnig of October. These attacks have resulted in a strongest homogeneity in the firm's communication but also on the frame structure.

(3) Finally, the contested process is fed back by an abductive fit with actions undertaken. A good example of this sub-process is the before-mentioned change that occured after some communicative clumsiness of France Telecom's CEO. Indeed, these speach acts have produced strong reactions inside public opinion and gave opponents an opportunity to advance their own statements. The major consequence was a "fit", a more adjusted and relevant way of communicate in newspaper and other media.

This study of the consteted process leads us to highlight the role of frame proponents and opponents. Through the development of their statements over time, we can identify several good examples of this process and its role in the dynamic of managerialisation.

Frame development through strategic intent. I now turn on another role of frame proponents. Indeed, beyond their statements, they can act directly on the framing process in order to enhance its efficiency. These attempts are four "frame-alignment strategies".

(1) Frame bridging strategy means the creation of a link between two or more structured and compatible frames around a particular issue. Here, the managerial frame is mainly built thanks to this strategy. Indeed, I found that the infancy managerial frame of workplace suicides was progressively fueled with another managerial frame: stress management intervention. A key indicator of this process is the shift of words and approach to solve the work related suicides issue. This shift happens in 4 moves. First of all, in the begining of October, the managerial frame begins to appear clearly thanks to elements such as « new social contract », « measure », « action plan » or « stage balance » (e.g. Les Echos, October 1 st , 2009; Le Figaro, October 2, 2009). Secondly, in the same time, the CEO is convened by the Minister of Labor and Economics. New words appear alongside "suicides" to express the issue: « psychosocial risk », « malaise », etc. The CEO and his team change also their words in order to talk to the press: « I focus exclusively on human. If employees are happy, the entire organization will win in the end. The new France Telecom should be humanly performance if we want to continue to be economically efficient. » (Le Figaro, October 17, 2009). But the approach begins to change too. Beside several emergency measures, the CEO announces a large survey: « We send these days a questionnaire to all employees where they can point out what is wrong. Everyone will be able to express themselves. » (Le Figaro, October 17, 2009). Third step, the preliminary findings of this survey are given on October.

This main event is announced as follows: « The specialized firm in workplace hazards must now submit its initial findings on stress at France Telecom. » (Les Echos, December 14, 2009). The issue moved from work related suicides to work related stress. This is confirmed in a last move in february 2010: « A major announcement in the negotiations on stress: two agreements, one on work-life balance, the other on mobility will be submitted for signatures to trade unions between february 19 and march 5. These negotiations began in September after the wave of suicides of employees and should lead to a new "social contract" in the company. » (Les Echos, February 10, 2010). The suicide issue is now totally integrated in a more general issue: work related stress. Although, this stress issue is also framed in a managerial way. Indeed, stress management interventions are part of a stable frame used to manage mental health issues related to work. Several managerial frameworks have been studied but they all follow the same logic: intervention preparation, stress assessment, actions plan, and efficiency assessment. As Harvey et al. (2006:13) put it: « These frameworks are largely similar to the change management models frequently referenced in the literature on organizational change; but here they have been adapted to the cause of stress intervention. » (for its main characteristics, see for example [START_REF] Brun | Strategic approach to preventing occupational stress[END_REF][START_REF] Cox | Research on work-related stress[END_REF][START_REF] Cooper | Healthy mind, healthy organization: A proactive approach to occupational stress[END_REF][START_REF] Harvey | Organizational Interventions and Mental Health in the Workplace: A Synthesis of International Approaches[END_REF]. We can say that this second mature managerial frame is mobilised to structure the first one. This is a typical frame bridging strategy.

(2) Frame amplification strategy means to highlight values and beliefs of the frame which are found in dominant culture's values and beliefs. In my case study, this is achieved through the use of crisis communication patterns. Indeed, in an exemplary interview named « We will emerge from the crisis together. » (Le Figaro, October 17, 2009), the CEO of France Telecom is trying to diffuse several important values when he talks about a new suicide: respect, humility and dignity (« I am appalled. Extremely sorry. [...] This news struck me as a new blow. All my thoughts are with the family we try to help as we always do in these very painful situations. »), personnal responsability and engagement but also humanity or familylike values (« Despite the crisis we are facing, [employees] work, the company is running.

But this visceral attachment also produces some expectations from the company, which becomes like a big family where everything takes an emotional turn. »). He calls also for solidarity and a more peaceful mood (« Trade unions also have a real commitment to the company. They must be responsible in the negotiations. They achieve their mission. ») and he signals its official support (« [the] Labor Minister is supporting us »). These quotes are representative of this frame strategy which aims to recruit new proponents and to legitimize its approach.

(3) Frame extension means the extention of the range of issues or problems targeted in a frame. In this case study, I didn't find piece of evidence of this process. This is probably due to the short period studied. Indeed, the frame is relatively new and had not the time to really evolve in order to target other topics. It could also be the influence of the crisis communication plan in which frame proponents appear coherent between themselves in order to preserve the firm's reputation. Discrepancies between proponents would have given a poorest image of a firm stricken by workplace suicides.

(4) Frame transformation means changing radically the underlying meanings of the frame in order to replace them by new ones more attractive. In this case study, this process could be considered as the first which appear and contribute strongly to creat the managerial one. Indeed, my findings indicate that a frame transformation occured from an original frame which is mainly based on the "individual weakness" (see first case study). However, this frame is fastly left for the managerial one. Pieces of evidence of this transformation are found in the evolution of France Telecom CEO's discourses. In the end of september, 2009, he mentionned several times that suicides would come from individual weakness (« I did not want to enter in a system where communication could dampen the moral of people who are weak by nature » Libération, september 25, 2009; « a free helpline will be set up for the weakest employess » Les Echos, septembre 25,2009). He also compared the repeated suicides as a fashion (« the reason why I communicate today is that we have to stop this suicide fashion » Libération, september 25, 2009). As I mentionned above (see contested process), the CEO was strongly criticized for his attitude and convened by Ministers. Since he has changed his communication strategy in the beginning of October, 2009, he also has changed the underlying frame used to think about, and to act upon, work related suicides.

Frame diffusion. The last framing process characteristic is its diffusion through mass media and the role of active adopters. The managerial frame studied here is largely diffused through mass media communication (105 articles in four newspapers in 9 months). This communication is firstly pulled by articles which reveal workplace suicides on July and august, 2009, or by a press conference organized on September 15, 2009, by the french Minister of Labor. Then, frame proponents engage themselves in volontary communicative actions. In the beginning of October, under a political and social pressure, France Telecom creates a crisis-cell thanks to a specialised consulting firm (which is Euro RSCG & Co). This crisis-cell produces a "media plan" which is implemented in the following days by the CEO both on press and radio broadcasts (Les Echos, October 15, 2009).

As my findings show, the managerial frame named "psychosocial risks management" is present in this second case study. It appears through a strategy of frame transformation motivated by a contested process. The CEO and its Executive Managers left their first approach for a relevant one, infused by managerial characteristics. Then, discursive and contested processes are both combined to give the frame its general aspect. It is also enriched just a few) view of the world, sometimes fragmented and confusing. I don't know how deep this managerialization of work related suicide is anchored in other areas of our society, e.g. in public opinion. Other case studies, based on multiple levels of analysis and data sources could help us to refine our findings. Futhermore, it will decrease the selection bias we face inevitably in newspaper (journalists or experts are generally selected -we don't get everybody's opinion). As a conclusion, I call for new studies in order to confirm or prolongate this first attempt to establish the key patterns of the managerialist dynamic.

1 All the data collection and analysis was initially conducted in french.

2 15 articles coming from Le Figaro, 25 articles coming from Le Monde, 16 articles coming from Les Echos, and 15 articles coming from Libération.

3 Quotes from newspapers' articles are translated from french by the author.


Frédéric Meunier
email: frederic.meunier@cermics.enpc.fr

Antoine A Deza
email: deza@mcmaster.ca

A further generalization of the colourful Carathéodory theorem

Keywords: Mathematics Subject Classification. 52C45, 52A35 Colourful Carathéodory theorem, colourful simplicial depth, discrete geometry

COLOURFUL CARATHÉODORY THEOREMS

Given d + 1 sets, or colours, S 1 , S 2 , . . . , S d +1 of points in R d , we call a set of points drawn from the S i 's colourful if it contains at most one point from each S i . A colourful simplex is the convex hull of a colourful set S, and a colourful set of d points which misses S i is called an i -transversal. The colourful Carathéodory Theorem 1.1 by Bárány provides a sufficient condition for the existence of a colourful simplex containing the origin 0. Theorem 1.1 [START_REF] Bárány | A generalization of Carathéodory's theorem[END_REF]). Let S 1 , S 2 , . . . , S d +1 be finite sets of points in R d such that 0 ∈ conv(S i ) for i = 1 . . . d + 1. Then there exists a set S ⊂ i S i such that |S ∩ S i | = 1 for i = 1, . . . , d + 1 and 0 ∈ conv(S). Theorem 1.1 was strengthened by Arocha et al. [ABB + 09] and by Holmsen et al. [START_REF] Holmsen | Points surrounding the origin[END_REF] who provided a more general sufficient condition for the existence of a colourful simplex containing the origin.

Theorem 1.2 ([ABB + 09, HPT08]). Let S 1 , S 2 , . . . , S d +1 be finite sets of points in R d such that 0 ∈ conv(S i ∪ S j ) for 1 ≤ i < j ≤ d + 1. Then there exists a set S ⊂ i S i such that |S ∩ S i | = 1 for i = 1, . . . , d + 1 and 0 ∈ conv(S).

We strengthen Theorem 1.2 by further generalizing the sufficient condition for the existence of a colourful simplex containing the origin. Moreover, the proof, given in Section 4, provides a short and geometric alternative proof for Theorem 1.2. Let --→ x k 0 denote the ray originating from x k towards 0. Theorem 1.3. Let S 1 , S 2 , . . . , S d +1 be finite sets of points in R d . Assume that, for each 1 ≤ i < j ≤ d + 1, there exists k ∉ {i , j } such that, for all x k ∈ S k , the convex hull of S i ∪ S j intersects the ray --→ x k 0 in a point distinct from x k . Then there exists a set S ⊂ i S i such that |S ∩ S i | = 1 for i = 1, . . . , d + 1 and 0 ∈ conv(S). Up to an additional argument to handle degenerate configurations, Theorem 1.3 can be derived from slightly stronger Theorem 1.4 where H + (T i ) denotes, for any i -transversal T i , the open half-space defined by aff(T i ) and containing 0. The proof of Theorem 1.4 and the argument to handle degeneracy for Theorem 1.3 are given in Section 4.

Theorem 1.4. Let S 1 , S 2 , . . . , S d +1 be finite sets of points in R d such that the points in i S i ∪{0} are distinct and in general position. Assume that, for any i = j , (S i ∪ S j ) ∩ H + (T i ) = for any i -transversal T i . Then there exists a set S ⊂ i S i such that |S ∩ S i | = 1 for i = 1, . . . , d + 1 and 0 ∈ conv(S).

Note that, as the conditions of Theorems 1.1 and 1.2, but unlike the one of Theorem 1.4, the condition of Theorem 1.3 is computationally easy to check. In the plane and assuming general position, Theorem 1.3 can be strengthened to Theorem 1.5, see Section 6 for a proof.

Theorem 1.5. Let S 1 , S 2 , S 3 be finite sets of points in R 2 such that the points in S 1 ∪S 2 ∪S 3 ∪{0} are distinct and in general position. Assume that, for pairwise distinct i , j , k ∈ {1, 2, 3}, the convex hull of S i ∪ S j intersects the line aff(x k , 0). Then there exists a set S ⊂ S 1 ∪ S 2 ∪ S 3 such that |S ∩ S i | = 1 for i = 1, 2, 3 and 0 ∈ conv(S).

Figures 1 and2 illustrate sets satisfying the condition of Theorem 1.3 but not the ones of Theorems 1.1 and 1.2. The d -dimensional version of the set given in Figure 1 corresponds to the points in S i being clustered around the i t h vertex of a simplex containing 0. Note that while all the (d + 1) d +1 colourful simplices of this configuration contain 0, for d ≥ 3 it does not satisfy the condition of Theorem 1.3 but does satisfy the one of Theorem 1.4. 0 FIGURE 2. A set in dimension 3 satisfying the condition of Theorem 1.3 but not the one of Theorem 1.2. 0 FIGURE 3. A set in dimension 3 satisfying, up to a slight perturbation, the condition of Theorem 1.4 but not the one of Theorem 1.3. Remark 1.6. By a standard perturbation argument, Theorem 1.4 still holds if the general position assumption is replaced by the following milder condition: for each transversal T i , there exists an hyperplane containing T i but not 0. Indeed, for each transversal T i , denote by H 0 (T i ) an hyperplane containing T i and not 0. Up to a perturbation of the points in i S i and the H 0 (T i ) accordingly, the configuration is in general position. With a sufficiently small perturbation, Theorem 1.4 can be applied because a point of i S i ∪ {0} not in H 0 (T i ) remains on the same side of H 0 (T i ). The colourful simplex containing 0 given by Theorem 1.4 did contain 0 before being perturbed since 0 is in its interior. Similarly, Theorem 1.5 holds if 0 is not aligned with two points of distinct colours.

For example, Remark 1.6 implies the existence of a colourful simplex containing 0 for the set given in Figure 3 while the condition of Theorem 1.3 is not satisfied for i = and j = . Figure 4 illustrates a set satisfying the condition of Theorem 1.5 but not the one of Theorem 1.4. 0 FIGURE 4. A set in dimension 2 satisfying the condition of Theorem 1.5 but not the one of Theorem 1.4.

GIVEN ONE, FIND ANOTHER ONE

In this section we present results that can be directly derived from the Octahedron Lemma, see end of Section 4.1 for more details about this lemma. Our objective is twofold:

(i ) to stress in Corollary 2.2 that, in general, the existence of one colourful simplex containing 0 implies the existence of at least min i |S i | colourful simplices containing 0. In other words, any condition implying the existence of a colourful simplex containing 0 actually implies the existence of min i |S i | such simplices, and (i i ) to obtain in Section 5 an algorithm finding a colourful simplex for sets satisfying the condition of Theorem 1.4 based on the algorithmic proof of Proposition 2.1.

Proposition 2.1. Given d + 1 sets, or colours, S * 1 , S * 2 , . . . , S * d +1 of points in R d with |S * i | = 2 for i = 1, . . . , d + 1, if
there is a colourful simplex containing 0, then there is another colourful simplex containing 0.

Proof. Without loss of generality we assume that the points in i S * i ∪ {0} are distinct and in general position. Consider the graph G whose nodes are of three types: (i ) T 1 of cardinality

d + 2 with 0 ∈ conv(T 1 ), |T 1 ∩ S * i | = 1 for i = 1, . . . , d , and |T 1 ∩ S * d +1 | = 2; (i i ) T 2 of cardinality d +1 with 0 ∈ conv(T 2 ), |T 2 ∩S * i | = 1 for i = 1, .
. . , d except for exactly one i , and |T 2 ∩S * d +1 | = 2; and (i i i ) T 3 of cardinality d + 1 with 0 ∈ conv(T 3 ) and |T 3 ∩ S * i | = 1 for i = 1, . . . , d + 1. The adjacency between the nodes of G is defined as follows. There is no edge between nodes of type T 2 and T 3 . A pair (T 1 , T 2 ), respectively (T 1 , T 3 ), is adjacent if and only if T 2 ⊆ T 1 , respectively T 3 ⊆ T 1 .

We now check the degree of T 1 , T 2 and T 3 nodes. First consider a T 1 node. We recall that, under the general position assumption, there are exactly two subsets X and Y of cardinality d + 1 containing 0 in their convex hull. Using the simplex method terminology, there is a unique leaving variable in a pivot step of the simplex method assuming non-degeneracy. Both of X and Y intersect S * i for i = 1, . . . , d in at least one point except maybe for one i . Thus, X and Y are T 2 or T 3 nodes, hence the degree of a T 1 node is 2. Consider now a T 2 node, there is a

i 0 = d + 1 such that |T 2 ∩ S * i 0 | = 0.
The T 2 node is contained in exactly two T 1 nodes, each of them obtained by adding one of the points in S * i 0 . Hence the degree of a T 2 node is 2. Finally, consider a T 3 node, it is contained exactly one T 1 node obtained by adding the missing point of S * d +1 . Hence, the degree of a T 3 node is 1. The graph G is thus a collection of node disjoint paths. Therefore, the existence of a colourful simplex containing 0 provides a T 3 node, and following the path in G until reaching the other endpoint provides another node of degree 1, i.e. a T 3 node corresponding to a distinct colourful simplex containing 0.

Corollary 2.2. Given d +1 sets, or colours, S 1 , S 2 , . . . , S d +1 of points in R d , if there is a colourful simplex containing 0, then there is at least min i |S i | colourful simplices containing 0.

Proof. Let I = min i |S i | and S i = {v 1 i , v 2 i , .
. . }, and assume without loss of generality that the given colourful simplex containing 0 in its convex hull is conv(v 1 1 , v 1 2 , . . . , v 1 d +1 ). Applying Proposition 2.1 (I -1) times with S * i = {v 1 i , v k i } we obtain an additional distinct colourful simplex containing 0 for each k = 2, . . . , I . Proposition 2.1 triggers the following problem, which we call SECOND COVERING COLOUR-FUL SIMPLEX. Given d + 1 sets, or colours, S 1 , S 2 , . . . , S d +1 of points in R d with |S i | ≥ 2 for i = 1, . . . , d + 1, and a colourful set S ⊂ i S i containing 0 in its convex hull, find another such set.

This problem is similar to the problem of, given one Hamiltonian circuit in a cubic graph, finding another one [START_REF] Papadimitriou | On the complexity of the parity argument and other inefficient proofs of existence[END_REF][START_REF] Andrew | Hamiltonian cycles and uniquely edge colourable graphs[END_REF], or to the room partitioning problem [START_REF] Edmonds | On finding another room-partitioning of the vertices[END_REF]. These problems are such that the existence of an object of a certain type implies the existence of another object of same type. The basic approach for such problems consists in representing the objects by nodes of odd degrees in a graph. Since there is always an even number of odd degree nodes in a graph, the existence of one odd degree node implies the existence of another one. It makes these problems belong to the PPA class defined by Papadimitriou [START_REF] Papadimitriou | On the complexity of the parity argument and other inefficient proofs of existence[END_REF] as search problems for which every instance is guaranteed to have a solution and for which the existence is proven through the following argument: In any graph with one odd degree node, there must be another odd degree node. Examples of PPA problems include BROUWER, BORSUK-ULAM, SECOND HAMILTONIAN CIRCUIT, NASH, see [START_REF] Papadimitriou | On the complexity of the parity argument and other inefficient proofs of existence[END_REF].

The PPA class has a subclass of PPA-complete problems for which the existence of a polynomial algorithm would imply the existence of a polynomial algorithm for any problem in complex defined by:

M = {F ∪ F : F ⊆ V (σ), F ⊆ V (σ ) and c(F ) ∩ c(F ) = }
where V (σ) denotes the vertex set of σ and c(x) = i for x ∈ S i . The simplicial complex M is a combinatorial (d -1)-pseudomanifold. Note that V (σ) and V (σ ) are abstract simplices of M . Let M be the collection of the convex hulls of the abstract simplices of M . Note that the vertices of all maximal simplices of M form d + 1-transversals and that M is not necessarily a simplicial complex in the geometric meaning as some pairs of geometric (d -1)-simplices might have intersecting interiors.

We recall that for any generic ray originating from 0, the parity of the number of times its intersects M is the same. We remark that this number can not be even as, otherwise, we would have a colourful (d -1)-simplex closer to 0 than σ on r since, M being contained in the closure of H + (T d +1 ), when r intersects σ, it is the last intersection. Thus, the number of times r intersects M is odd, and actually equal to 1. Take now any point v ∈ S d +1 and consider the ray originating from 0 towards the direction opposite to v. This ray intersects M in a colourful (d -1)-simplex τ; that is, 0 ∈ conv(τ ∪ {v}).

Assuming i S i lies on the sphere S d -1 , note that i -transversals generate full dimensional colourful cones pointed at 0. We say that a transversal covers a point if the point is contained in the associated cone. Colourful simplices containing 0 are generated whenever the antipode of a point of colour i is covered by an i -transversal. In particular, one can consider a combinatorial octahedra, or cross polytopes, generated by pairs of disjoint i -transversals, and rely on the fact that every octahedron Ω either covers all of S d -1 with colourful cones, or, every point x ∈ S d -1 that is covered by colourful cones from Ω is covered by at least two distinct such cones. In the case where the points of Ω form an octahedron in the geometric sense, these correspond to the cases where 0 is inside and outside Ω respectively. For a proof, see for example the Octahedron Lemma of [START_REF] Bárány | Quadratically many colorful simplices[END_REF]. The proof of Theorem 1.4 can therefore be reformulated as: (i ) either the pair of d + 1-transversals (T, T ) forms a octahedron covering S d -1 , or (i i ) 0 belongs to a colourful simplex having conv(T ) as a facet.

Corollary 4.1. Given d + 1 sets, or colours, S 1 , S 2 , . . . , S d +1 of points in R d , if there is a pair of transversals forming a octahedron covering S d -1 , then there is at least

min i = j |S i ∪ S j | -2 colourful simplices containing 0. Proof. Denote S i = {v 1 i , v 2 i , . . . }.
Without loss of generality, we can assume that the octahedron covering S d -1 is formed by a pair (T 1 , T 2 ) of d + 1-transversals such that 4.2. Argument to handle degeneracy for Theorem 1.3. As noticed in Remark 1.6, Theorem 1.4, and therefore heorem 1.3, holds if, for each transversal T i , there exists an hyperplane containing T i but not 0. Actually, one can check that the proof of Theorem 1.4 still works if there is at least one transversal T such that 0 ∉ aff(T ) and such that the points of T are affinely independent. Indeed, we can choose a ray r such that the non-empty intersections with aff(T ) for any T are distinct. Without loss of generality, we choose a transversal T d +1 = {v 1 , . . . , v d } in the proof such that aff(T d +1 ) is the first affine subspace spanned by a transversal intersected by r. Then, setting j = d + 1, the condition of Theorem 1.3, namely the existence of a k ∈ {1, . . . , d }\{i } such that the convex hull of S i ∪S d +1 intersects the ray

T 1 = {v 1 1 , v 1 2 , . . . , v 1 d } covers a point in S d +1 . Applying Proposition 2.1 (I -2) times with S * i = {v 1 i , v k i }
--→ v k 0 in a point distinct from v k , implies (S i ∪ S d +1 ) ∩ H + (T d +1 ) = .
Therefore, an additional argument is needed only to handle the case when the maximum cardinality a of an affinely independent colourful set whose affine hull does not contain 0 is at most d -1. We can choose a ray r such that the non-empty intersections with aff(A) for any colourful set A of cardinality a are distinct. Let A 0 be the colourful set of cardinality a such that aff(A 0 ) is the first intersected by r. Without loss of generality, let A 0 be {v 1 , . . . , v a } with v s ∈ S s . Note that S a+1 ∪. . .∪S d +1 ⊂ aff(A 0 ∪{0}) as otherwise 0 ∉ aff(A 0 ∪{v j }) for v j ∈ S j with j > a, which contradicts the maximality of a. If there is a colourful simplex containing 0, we are done. Therefore, we can assume that, in aff(A 0 ∪ {0}), 0 is in the open half-space defined by aff(A 0 ) and not containing S a+1 ∪ . . . ∪ S d +1 , and will derive a contradiction.

Let B 0 = {a + 1, . . . , d + 1} and note that |B 0 | ≥ 3. We remark that, for all i , j ∈ B 0 with i = j , the k, such that conv(S i ∪ S j ) intersects --→ x k 0 in a point distinct from x k , satisfies k ∈ B 0 , since S i ∪ S j are separated from 0 by aff(A 0 ) in aff(A 0 ∪ {0}). We can now define the following set map:

F (B ) = {k : ∃(i , j ) ∈ B × B, i = j such that conv(S i ∪ S j ) ∩ --→ x k 0 {x k }} if |B | ≥ 2 otherwise.
We have

F (B ) ⊂ F (B ) if B ⊂ B . Let B = F (B -

ALGORITHM TO FIND A COLOURFUL SIMPLEX

We present an algorithm based on the proof of Proposition 2.1 finding a colourful simplex containing 0 for sets satisfying the conditions of Theorem 1.4, and, therefore, for sets satisfying the condition of Theorem 1.3.

Algo1. Take any colourful (d -1)-simplex σ whose vertices form, without loss of generality, a d + 1-transversal T = {v 1 , . . . , v d }, and a ray r intersecting σ in its interior. Let H + (T ) be the open half-space delimited by aff(T ) and containing 0. Check if there is a colourful d -simplex having σ as a facet and either containing 0 or having a facet τ intersecting r before σ. If there is none, we obtain d new vertices forming a d + 1-transversal T = {v 1 , . . . , v d }, see Section 4. Take a point x on r with x ∉ i S i and 0 ∈ conv(v 1 , . . . , v d , x) and any point v d +1 ∈ S d +1 . We can use Proposition 2.1 and its constructive proof with S i = {v i , v i } for i = 1, . . . , d and S d +1 = {x, v d +1 } to obtain a new colourful simplex containing 0 with at least one vertex in T . If v d +1 is a vertex of the new simplex, we do have a colourful simplex containing 0. Otherwise, the facet of the simplex not containing x is a colourful (d -1)-simplex τ intersecting r before σ since aff(T ) forms the boundary of H + (T ).

In other words, given a colourful (d -1)-simplex σ intersecting r, Algo1 finds either a colourful simplex containing 0, or a colourful (d -1)-simplex τ intersecting r before σ. Since there is a finite number of colourful (d -1)-simplices, the algorithm eventually finds a colourful simplex containing 0.

PROOF OF THEOREM 1.5 IN THE PLANE USING GRAPHS

In this section we present a proof of Theorem 1.5 for the planar case which also provides a alternative and possibly more combinatorial proof of Theorem 1.4 in the plane. We assume without loss of generality that S 1 ∪ S 2 ∪ S 3 ∪ {0} are distinct and in general position. Consider the graph G = (V, E ) with V = S 1 ∪ S 2 ∪ S 3 and where a pair of nodes are adjacent if and only if they have different colours. We get a directed graph D = (V, A) by orienting the edges of G such that 0 is always in the right side of any arc. Since conv(S i ∪S j )∩aff(x k , 0) = with i , j , k pairwise distinct and x k ∈ S k , we have deg + (v) ≥ 1 and deg -(v) ≥ 1 for all v ∈ V . It implies that there exists at least one circuit in D, and we consider the shortest circuit C . We first show that the length of C is at most 4 since any circuit of length 5 or more has necessarily a chord. Indeed, take a vertex v, there is a vertex u on the circuit at distance 2 or 3 having a colour distinct from the colour of v, and thus the arc (u, v) or (v, u) exists in D. Therefore, the length of C must be 3 or 4. If the length is 3, we are done as the 3 vertices of C form a colourful triangle containing 0. If the length is 4, the circuit C is 2-coloured as otherwise we could again find a chord. Consider such a 2-coloured circuit C of length 4 and take any generic ray originating from 0. We recall that given an oriented closed curve C in the plane, with k + , respectively k -, denoting the number of times a generic ray intersects C while entering by the right, respectively left, side, the quantity k + -k -does not depend on the ray. Considering the realization of C as a curve C , we have k -= 0 by definition of the orientation of the arcs. Since we can choose a ray intersecting at least once C , k + remains constant and non-zero. Take now a vertex w of the missing colour, and take the ray originating from 0 in the opposite direction. This ray intersects an arc of C since k + = 0, and the endpoints of the arc together with w form a colourful triangle containing 0.

0FIGURE 1

 1 FIGURE 1. A set in dimension 2 satisfying the condition of Theorem 1.3 but not the one of Theorems 1.2.

  we obtain, for each k = 3, . . . , I , an additional distinct colourful simplex containing 0 distinct from the |S d +1 | colourful simplices containing 0 given by the assumption that (T 1 , T 2 ) covers S d -1 .As noted in[START_REF] Deza | More colourful simplices[END_REF], the condition i conv(S i ) = implies the existence of at least one octahedron covering S d -1 . Thus, setting |S i | = d + 1, Corollary 4.1 yields µ(d ) ≥ 2d as proven in[START_REF] Deza | Colourful simplicial depth[END_REF].

  1 ) for = 1, 2, . . . As remarked above B 1 ⊂ B 0 , and, by induction, B ⊂ B -1 for all . The sequence (B ) converges towards a set B * satisfying F (B * ) = B * . Finally, note that, by induction, |B | ≥ 3: The base case holds as |B 0 | ≥ 3, and a pair i , j ∈ B with i = j yields a k ∈ B +1 , then i , k yields an additional k in B +1 , which in turn, with k, yields a third element in B +1 ; and thus |B * | ≥ 3. For any v ∈ k∈B * S k , the ray -→ v0 intersects the convex hull of k∈B * S k in a point distinct from v since F (B * ) = B * . It contradicts the fact that aff(A 0 ) separates 0 from S a+1 ∪. . .∪S d +1 in aff(A 0 ∪ {0}) by the following argument. There exists at least one facet of conv( k∈B * S k ) whose supporting hyperplane separates 0 from conv( k∈B * S k ) and, for a vertex v of this facet, -→ v0 ∩ conv( k∈B * S k ) = {v}, which is impossible.

Acknowledgments This work was supported by grants from the Natural Sciences and Engineering Research Council of Canada (NSERC), MITACS, and Fondation Sciences Mathématiques de Paris, and by the Canada Research Chairs program. We are very grateful to Sylvain Sorin and Michel Pocchiola for providing the environment that nurtured this work from the beginning.

PPA. The first example of a PPA-complete problem was given by Grigni [START_REF] Grigni | A Sperner lemma complete for PPA[END_REF]. We do not know whether SECOND COVERING COLOURFUL SIMPLEX is PPA-complete.

MINIMUM NUMBER OF COLOURFUL SIMPLICES CONTAINING 0

The minimum number µ(d ) of colourful simplices containing 0 for sets satisfying the condition of Theorem 1.1 was investigated in [START_REF] Deza | Colourful simplicial depth[END_REF]. This paper shows that 2d ≤ µ(d ) ≤ d 2 + 1, that µ(d ) is even for odd d , and that µ(2) = 5, and conjectures that µ(d ) = d 2 + 1 for all d ≥ 1. Subsequently, [START_REF] Bárány | Quadratically many colorful simplices[END_REF] verified the conjecture for d = 3 and provided a lower bound

, before [START_REF] Deza | More colourful simplices[END_REF] showed that µ(d ) ≥ (d +1) 2 2 . Note that since the points i S i ∪ {0} are distinct and in general position, 0 ∈ conv(S i ) implies |S i | ≥ d + 1. Therefore, one can consider the analogous quantity µ (d ), respectively µ • (d ), defined as the the minimum number of colourful simplices containing 0 for sets satisfying the condition of Theorem 1.2, respectively Theorem 1.4, and We recall that a k-simplex σ is the convex hull of affinely independent (k + 1)-points. An abstract simplicial complex is a family F of subsets of a finite ground set such that whenever F ∈ F and G ⊆ F , then G ∈ F . These subsets are called abstract simplices. The dimension of an abstract simplex is its cardinality minus one. The dimension of a simplicial complex is the dimension of largest simplices. A pure abstract simplicial complex is a simplicial complex whose maximal simplices have all the same dimension. A combinatorial d -pseudomanifold M is a pure abstract d -dimensional simplicial complex such that any abstract (d -1)-simplex is contained in exactly 2 abstract d -simplices.

Consider a ray r originating from 0 and intersecting at least one colourful (d -1)-simplex. Under the general position assumption for points in i S i ∪ {0}, one can choose r such that it intersects the interior of the colourful (d -1)-simplex, and that no two colourful simplices have the same intersection with r. Let σ be the first colourful (d -1)-simplex intersected by r. Note that, given r, σ is uniquely defined. Without loss of generality, we can assume that the vertices of σ form the d + 1-transversal {v 1 , . . . , v d }.

Setting j = d +1, and

In other words, there is, for each i , a point either in

Assume first that for one i the corresponding point belongs to S d +1 , and name it v d +1 . Then r intersects the boundary of conv(v 1 , . . . , v d , v d +1 ) in only one point as otherwise r would intersect another colourful (d -1)-simplex before intersecting σ. This is impossible since r leaves H + (T d +1 ) after intersecting σ. Thus, r intersects conv(v 1 , . . . , v d , v d +1 ) in exactly one point; that is, 0 ∈ conv(v 1 , . . . , v d , v d +1 ). Therefore, we can assume that for each i there is a point v i = v i in S i ∩ H + (T d +1 ), and consider the d + 1-transversal T = {v 1 , . . . , v d } and the associated colourful (d -1)-simplex σ = conv(v 1 , . . . , v d ). Let M be the abstract simplicial