Louis Bertrand

Vincent Caradec

Jean-Sébastien Eideliman

Situating Disability. The recognition of "handicapped workers" in France Le handicap en situation

Keywords: disability, situation of handicap, employment, RQTH, life course Résumé

construire les travailleurs handicapés comme une sous-catégorie des personnes handicapées, le positionnement est plus ambivalent du côté des demandeurs. Avec leurs mots, beaucoup utilisent de fait la notion de situation de handicap, très présente dans les débats actuels sur le handicap mais non retenue par la loi française de 2005 : s"ils sont gênés dans certaines situations professionnelles, beaucoup ne se pensent pas pour autant comme des personnes handicapées. Pour comprendre ces décalages, nous proposons d"une part de revenir aux fondements de la politique française d"emploi en faveur des personnes handicapées, d"autre part d"explorer la variété des enjeux qui président aux demandes de reconnaissance, sans oublier le rôle d"interface essentiel des professionnels du champ du handicap. Notre réflexion s"appuie sur une enquête de terrain qui s"est déroulée des deux côtés du « guichet » des Maisons départementales des personnes handicapées, auprès d"agents de l"institution et de demandeurs d"une reconnaissance de la qualité de travailleur handicapé.

Mots-clefs : handicap ; situation de handicap ; emploi ; RQTH ; parcours de vie As recent as the notion of "handicap" may be in France, only really appearing in the late 1950s, it already has a long and turbulent history. 1 "Handicap" (as disability is still known in France) was the latest in a long series of terms chosen to escape the negative connotations that had gradually came to taint preceding expressions (infirmity, abnormality, maladjustment…) [START_REF] Stiker | A History of Disability[END_REF]. It came to define a new field of social policy when a pair of laws 2 in 1975 tried to harmonize the private initiatives that had until then dominated the field of disability, bring them under centralized State control [START_REF] Ebersold | L'invention du handicap: la normalisation de l'infirme[END_REF]. Thirty years later, a 2005 law 3 updated the principles of the sector and gave the first legal definition of "handicap." 4 The fruit of intense negotiation, this law failed to embrace the notion of "person in a situation of handicap" (personne en situation de handicap; [START_REF] Winance | Being normally different? Changes to normalization processes: from alignment to work on the norm[END_REF], as promoters of the social model of disability in the wake of disability studies would have liked: it defined disability as the consequences of a "substantial, persistent, or permanent change" in physiological function. The emphasis is put on individual situations in two ways, however: first by specifying that the consequences of these changes are "suffered by a person in his environment," then by re-enforcing the process for evaluating applications case by case through the creation of the Maisons Départementales des Personnes Handicapées (Departmental Centres for Handicapped Persons; MDPH), developed as a centralised portal to the full range of claims and claimants. Within them, collegial groupings (multi-disciplinary teams and the Commission des droits et de l'autonomie des personnes handicapées 1 Loi n° 57-1223 du 23 novembre 1957 sur le reclassement professionnel des travailleurs handicapés (Law n° 57-1223 of 23 November 1957 on the professional reclassification of handicapped workers). 2 Loi n° 75-534 du 30 juin 1975 d'orientation en faveur des personnes handicapées and Loi n° 75-535 relative aux institutions sociales et médico-sociales (Law n° 75-534 of 30 June 1975 for orientation in favour of handicapped persons; Law n° 75-535 relative to social and medico-social institutions). 3 Loi n° 2005-102 du 11 février 2005 pour l'égalité des droits et des chances, la participation et la citoyenneté des personnes handicapées (Law n° 2005-102 of 11 February 2005 for the equality of rights and opportunities, the participation and citizenship of handicapped persons). 4 Article 2 holds that "A handicap, in the meaning of this present law, is any limitation on activity or restriction on participation in social life suffered by a person in his environment due to a substantial, persistent, or permanent alteration of one or several physical, sensorial, mental, cognitive, or psychic functions, a multiple handicap, or an invalidating health problem."

(Commission for the Rights and Autonomy of Handicapped Persons; CDAPH)) grant, when appropriate, rights, status, or orientations depending on explicitly stated requests and the needs and expectations that may be expressed in the application"s "Life Plan" page. The "Life Plan" might refer to issues that help contextualise the situation of disability in multidimensional living circumstances.

As we can see, the choices made in the 2005 law are situated among wider processes and a range of tensions: between the individualisation of social policy and mass bureaucratic processing in a context of cost reduction [START_REF] Dubois | The bureaucrat and the poor: Encounters in French Welfare Offices[END_REF]; between medical and social models for disability (Ville et al., 1994); between an integrative rationale aiming to compensate for disability by favouring people with formal handicap recognition and an inclusive rationale aiming to transform social frames to make them more adaptable [START_REF] Winance | Being normally different? Changes to normalization processes: from alignment to work on the norm[END_REF]. The 2005 law hedges its bets among these extremes more than it chooses sides. As we have shown elsewhere [START_REF] Bertrand | Saisir les personnes. Le fonctionnement des commissions dans l"aide au logement et aux personnes handicapées[END_REF], MDPHs function as "personalised bureaucracies" where power is slowly finding a new balance between the previously dominant medical pole and the strengthened social pole. This article"s overall aim is to contribute to a better understanding of the consequences of these transformations in how situations are taken into account in the field of disability.

We found the domain of employment to be particularly interesting for raising these questions. Historically, work is one of the sites where the notion of disability is elaborated, through a dual process of particular interest for the subject of our study: on one hand work may be seen as a place of rehabilitation where handicapped persons may find or re-find a feeling of social utility and self-realisation, so long as their work is appropriate for them; on the other hand, work may be a setting where situations of handicap appear, because performance demands might reveal or even engender limitations that then become problematic. For one, taking the example of the foundational figures of disability noted by Henri-Jacques [START_REF] Stiker | A History of Disability[END_REF], we could cite people with tuberculosis and disabled veterans for whom jobs were reserved or adapted following the first World War; for the other, there is the example of workers victim to accidents on the job (starting in 1898), or simply occupational wear and tear that partially or entirely disqualifies them on the employment market.

More specifically, the study of measures for the Recognition of the Quality of Handicapped Worker (Reconnaissance de la Qualité de Travailleur Handicapé; RQTH) will be the focus of our attention in this article. Attribution is based on Article L5213-1 of the Code du travail (the French labour code), which states: "considered to be a handicapped worker is any person whose possibilities for obtaining or keeping a job are effectively reduced following the alteration of one or several physical, sensorial, mental, or psychological functions." RQTH status gives access to certain forms of assistance, workstation accommodation in particular, and is a prerequisite for access to specialized vocational rehabilitation and job-placement centres: the Cap Emploi network, offering customized job-seeking services, and the SAMETH network (Services Helping Handicapped Workers Maintain Employment). Thus, according to the law, the notion of "handicapped worker" includes a range of figure-types, from the beneficiary of an institution or work aid centre (ESAT, formerly the CAT) to a white-collar worker with a disability in an ordinary work setting, and including the unemployed, people with work-related injuries, people changing careers following a disqualifying allergy, and so on. So which elements of each situation do commission members retain in granting "recognition of the quality of handicapped worker"? Are the same elements thought to be important by both the decisionmakers and claimants? And don"t these elements vary from one claimant to another?

To understand the issues behind the notion of "situation" for disabled workers under the 2005 law and how the notion has been used, this article uses findings from a field study the three authors conducted with RQTH claimants and decision-makers between 2010 and 2012 in two French administrative departments with different economic, social, and political contexts. We were able to conduct forty observations of teams working on applications (multi-disciplinary teams and CDAPH), fifteen interviews with agents working in or with the studied MDPH, and thirty-seven interviews with people applying for RQTH. 5We will begin with a presentation of the specificities of employment within disability policy by showing that this question occupies an ambiguous place in the French disability field: historically located at the core of public disability policy, it now has a place apart. We will then see that the professions who evaluate RQTH claims do consider certain aspects of the claimants" environments, but mainly try to recognize disabilities regardless of situations: the identification of disabled persons with particular employment-related needs takes priority over workers in situations of handicap arising from their working conditions. Lastly, on the claimants" side, their requests are motivated by other aspects of their personal situations, which vary widely depending on where they are in their life course. Generally speaking, many initially try to get arrangements in response to difficulties that they do not necessarily perceive as disabilities. Thus, in addition to the ensemble of handicapped persons trying to claim their right to employment, we find a large number of people who primarily see themselves as current or potential workers with particular needs that are mainly expressed as a variety of forms of requests for protection.

Methodology

The study was conducted in parallel in two French administrative departments with contrasting social, political, and economic characteristics. Since the two relevant MDPH agreed to participate in the study, we were able to observe the variety of multi-disciplinary teams that handle professional integration. We also observed meetings of both CDAPH, in plenary sessions and in small groups. In all, we attended twenty-seven multi-disciplinary team meetings and thirteen CDAPH meetings.

During this observation period, we asked some of the professional we met (doctors, social workers, employment counsellors, MDPH coordinators…) for an interview about their professional practices. Fifteen of them accepted (eight in one administrative department, seven in the other). In parallel, using information collected from applications, we chose a sample of people who had received an RQTH decision in the year 2010. The selected claimants were contacted by letter, then by telephone. We diversified our sample by taking care to include both people whose claims were granted and those who were refused, as well as people who had been granted or refused Handicapped Adult Benefits (Allocation Adulte Handicapé; AAH). At the end of the interview we asked their permission to consult their files; most interviewees accepted, which allowed us to better reconstruct the claim"s institutional progression and gave us access to the "Life Plan" (when there was one). In all, we interviewed seven people who were refused both the RQTH and AAH; sixteen who were granted the RQTH but not AAH, and fourteen people who were granted both (thirty-seven interviews in total).

The place names and names of individuals used in this article have been changed.

Moreover, everyone we questioned was informed of his or her right to refuse interviews or answering particular questions. Findings were protected, and all data made public has been anonymized and disseminated only in scientific settings. Lastly, we presented our research findings to the institutions that were the object of our study.

The French conception of disabled workers

Employment holds a special place in French disability-related policy. For one thing, it is at the very origins of the field of disability [START_REF] Stiker | A History of Disability[END_REF]. And yet the foundational laws of 1975 and 2005 contribute little new to the issue compared to those of 1957 and 1987, 6 which specifically addressed the employment question. So how do the general orientations of this young field of disability, which currently emphasizes accessibility, compensation, and the role of the surrounding environment, articulate with the more specific policy for the employment of disabled workers? Although the professional setting might be the privileged place for applying the notion of "situation of handicap," it is much more ambivalent in practice.

Work, the foundation of disability policy

Henri-Jacques [START_REF] Stiker | A History of Disability[END_REF] traces the origin of the French State"s assumption of responsibility in the disability field to three events that, in addition to being important steps in the construction of a welfare State, were powerful moments legitimizing national solidarity with populations weakened by events of origins professional (the 1898 work-related accidents law), military (the consequences of the first World War), or pathological (the late 19 th -early 20 th century tuberculosis epidemic). On each occasion, society experienced a rising feeling of national indebtedness toward people who had lost physical integrity for the common good (defence of the national territory; industrial production is responsible for both 6 Loi n° 87-517 du 10 juillet 1987 en faveur de l'emploi des travailleurs handicapés (Law n° 87-517 of 10 July 1987 in favour of the employment of handicapped workers).

for accidents and the constitution of a working class living in poor conditions propitious to epidemics). Each time, debates at least partly touched on both the right to work (via policies for re-integration, passed as early as 1924 with the institution of a disabled war veteran hiring quota for businesses) and on the right not to work (establishment of compensation, benefits, and pensions) [START_REF] Ville | From inaptitude for work to trial of the self: The vicissitudes of meanings of disability[END_REF].

Closely connected with employment issues from the outset, the field of disability was structured in consequence, at a time when life came to be increasingly ordered by institutions (school, the employment market, retirement) guiding and naturalizing a tripartite division of life into childhood, adulthood, and old age [START_REF] Kohli | The World We Forgot: A Historical Review of the Life Course[END_REF]. If the programmes devoted to socalled abnormal (to become "maladjusted") childhood and to disabled adults have merged into the field of disability, the boundary with the more aged has been maintained, as shown by the persistence of an eligibility threshold of 60 years of age for several measures, effectively distinguishing disability (in children and adults) from dependency (in the elderly) [START_REF] Weber | Handicap et dépendance. Drames humains, enjeux politiques[END_REF]. The consequences of these divisions by criteria of age are significant and not limited to administrative divisions; they also influence how individuals categorize their health problems [START_REF] Béliard | Mots pour maux. Théories diagnostiques et problèmes de santé[END_REF]. Jean-François Ravaud, Isabelle Ville and Alain Letourmy (2002) show that the probability of declaring oneself disabled, for equivalent everyday limitations, "depends on the economic value of a person in terms of employment" (Ibid., p. 549). All other things being equal, then, women and people not of a generally employed age declare themselves to be disabled less frequently. Employment thus seems to be at the heart of administrative as well as individual definitions of disability. It is moreover symptomatic that the first appearance of the term "handicapped" (in adjectival form) in law was in the 1957 law relative to the "reclassification of handicapped workers," meaning that, officially speaking, disabled workers existed before disabled persons did.

From right to work to obligation to employ

It is surprising to see to how little the 1975 and 2005 laws, which aimed to define and redefine the field of disability as a whole, addressed the employment question. For the most part, they merely took up provisions that had already been established under other laws, especially those of 1957 and 1987, at best re-enforcing them. Following a trend underway since the end of the First World War, these laws established an affirmative action program quite novel for French social policy, based on hiring quotas for the mutilated, infirm, and/or disabled workers in businesses. Subsequent laws would lower these quotas (from 10 to 6%) but extend them to the public sector and, most importantly, enforce them more tightly (Blanc, 2009). The legal employment rate for disabled workers, defined by a formula set in law, has been increasing steadily since 1987, and 2011 estimates put disabled workers at 5% of the workforce. However, the direct employment rate is closer to 3%, because many businesses respond to the "Obligation to Employ Handicapped Workers" (OETH) in other ways, such as making a financial contribution to a fund for workplace integration,7 sub-contracting with businesses in the sheltered workshop sector, or taking disabled interns in the name of professional training. 8Keeping such quotas is rather unique given the French social policy tradition, which is rarely in the register of affirmative action, and recent orientations in the United Kingdom [START_REF] Woodhams | Then and now: disability legislation and employers' practices in the UK[END_REF], Sweden [START_REF] Cohu | Les politiques en faveur des personnes handicapées dans cinq pays européens[END_REF], and the United States and Canada [START_REF] Oakes | Perspectives on Disability, Discrimination, Accomodations and Law. A comparison of the Canadian and American Experience[END_REF]. Despite some significant differences between these countries, recent disability legislation took inspiration from civil rights policy and implemented anti-discrimination systems that sometimes replaced quota-based affirmative action systems. At the same time, a fair number of European countriesnotably Germany, Austria, and Italystill have quotas similar to those in France [START_REF] Cohu | Les politiques en faveur des personnes handicapées dans cinq pays européens[END_REF].

The 1987 French law declares that the obligation to employ concerns people in at least one administratively recognized category of disability, 9 and nearly three quarters of the workers satisfying the OETH requirement have RQTH disabled worker status. 10 This measure is complex because it aims to bring together two issues and three actors: applicants, MDPH agents, and businesses must find a way for a proportion of the first group to find and/or keep suitable jobs, and for the last group to be able to count disabled workers on their staff to meet their obligation to employ. And yet a request for disabled worker status can only come from the concerned individuals, who are not required to tell their employers of their intention or ask for a renewal when recognition expires (after five years at the longest).

Although businesses have an interest in potentially eligible employees being recognized as disabled, the same does not always hold for individuals, for whom recognition of a disability is potentially both a resource and a stigma [START_REF] Goffman | Stigma: Notes on the management of spoiled identity[END_REF]. From business"s perspective, there is often considerable anxiety about hiring disabled workers, since it is difficult to anticipate the consequences of a disability on productivity and the working group. In order to meet the quota, then, it is less risky for them to try to get the MDPH to recognize alreadyhired workers as disabled. This is why large companies in particular may incite occupational physicians to encourage their employees to take the necessary steps. Some MDPH facilitate the process by accepting simplified applications for RQTH status coming directly from the 9 These categories are as follows: Recognition of the Quality of Handicapped Worker (RQTH); disability card holder; recipient of Handicapped Adult Benefits (AAH); victim of work-related accident or illness leading to permanent incapacitation of at least 10%; recipient of a disability pension due to a reduction of work capacity or earnings by at least two-thirds; disabled veteran or equivalent. For more detail, see http://travailemploi.gouv.fr/informations-pratiques,89/les-fiches-pratiques-du-droit-du,91/travailleurs-handicapes,1976/lobligation-d-emploi-en-faveur,12746.html. 10 DARES data. occupational medical service. Thus, policy for the employment of disabled workers develops in a way that is in many respects parallel to disability policy itself. Moreover, requesting RQTH from the MDPH does not lead to the attribution of a disability rating (which is given for many other disability statuses), meaning that someone may be recognized as a disabled worker without having either a formal disability rating or a card attesting to any disability. Would this mean that where employment is concerned, it is "situations of handicap" that are identified rather than "handicapped persons"? In fact, the following presentation of our study of the practical operations of MDPHs shows the reverse.

Maintaining disability outside of "situations"

Neither the 2005 law nor the history of handicapped worker recognition alone suffices for anticipating how the systems in place in France today actually work. The previously described tensions, between consideration of individual characteristics and elements specific to given situations, leave teams charged with granting handicapped worker status a nonnegligible flexibility. Although these practices obviously do not represent how the entire French field of disability works, they do show one way in which disability professionals position themselves. Over the course of our study, our observations of interdisciplinary teams specialized in employment-related issues thus allowed us to understand how they make RQTH attribution decisions in real-life cases. Despite the variety of procedures and the arbitration between departments, some patterns emerge that permit greater understanding of how policy for the employment of disabled persons works in practice. We will begin with a summary of distinctive aspects of MDPH operations before demonstrating some of our results with two observed sequences that are congruent with the rest of our material (observations and interviews with professionals).

A personalized bureaucracy

Disability-related application files travel far between their submission and the final decision, variable according to the application"s complexity and each MDPH"s procedures. In some departments, all or most applications go to a doctor who evaluates the claim, but in other departments they are quite often subject to an interdisciplinary group examination.

Employment-related requests may be examined by specialized interdisciplinary teams, which necessitate bringing in professionals from a variety of partner institutions, such as Cap Emploi, Pôle Emploi (the national job placement service), CARSAT (the insurance fund for retirement and work-related health), vocational training centres, and the like. Since the 1980s, social policy programmes have been urged to focus their efforts on "situations" or "persons," as opposed to "cases" or "files" [START_REF] Bertrand | Situations et problématiques: le travail d"une magistrature sociale par ses mots[END_REF], with the idea of personalizing social assistance, but a great many decisions are still taken routinely, based on limited criteria and data. Furthermore, RQTH claims have a particularly high positive response rate (on the order of 95%) that is relatively stable from year to year.

Our observations prompted us to relativize opposition between individualized social policy and the bureaucracy [START_REF] Bertrand | Saisir les personnes. Le fonctionnement des commissions dans l"aide au logement et aux personnes handicapées[END_REF], following other studies such as that of Vincent Dubois ( 2010). Yet insistence on taking account of individuals is not simply a ruse, and it leads to significant transformations in administrative work and the relationship with users. The Commissions for the Rights and Autonomy of Handicapped Persons (CDAPH), where official decisions are made following their processing by multidisciplinary teams, present themselves as the showcase for this new ideal because they discuss a small fraction of the application files case by case. This procedure does not reflect the everyday reality of work in the MDPH, however, where personalization still means bureaucratization. To use Max Weber"s term (1958)although in near opposition to his own analysis -personalization contributes to efforts to "re-enchant" bureaucratic relationships, which change much more slowly than the frequency of legal and organizational revisions might lead one to expect.

Handling situations, understanding individuals

In these commissions (either multidisciplinary or CDAPH), professionals often speak of "situations," which they are supposed to imagine holistically by taking account of the applicant"s "Life Plan," if he or she chose to draft one (they are optional in the application).

But they also have to conform to rules having the essential function of assuring that claims are handled as fairly as possible. In practice, the aim of the global evaluation of "the situation" is generally overshadowed by a more established attitude found in French social policy, consisting of attributing rights according to inherent characteristics of the applicant. This is especially clear in discussions on the possibility of retracting benefits from a person whose situation has changed since his or her previous claim. In the following case, the situation was further complicated by a reform introduced by the 2009 finance law, instructing MDPHs to consider the possibility of granting RQTH to people who had in fact requested the more exclusive Handicapped Adult Benefits (AAH). 11Mr Bécaria was born in 1962. 12 A specialized interdisciplinary team re-examined his file in July 2010, when his previous claim for AAH had been answered with an offer of RQTH. The file was presented as follows13 :

-MDPH branch coordinator (female) [Coord]: He was asking for an AAH renewal, but he was offered an SAP [project-support service14 ] and RQTH. He wrote us a letter complaining, two letters even… (quoting): "Go earn your money someplace else!" (laughs) -MDPH job placement specialist (female) [JPS]: (addressing the sociologist) That"s a case for you! It"s because of the reform.

-Social worker (male) [SW]: He deserves AAH.

-JPS: He was given a [disability] rating below 50% the first time, in 1996. He made an appeal, which was rejected because he didn"t show up. In 2002 he was set at 50% and AAH for a year, to help him re-integrate. -JPS: He also has a depressive syndrome and social problems.

-Coord: What do we do? -JPS: He hasn"t followed up on the suggestion to go see SAP, he doesn"t want RQTH, he hasn"t had a file at Cap Emploi since 2004. We"ve got to declare AAH renewal for two years. 15 We keep a rating between 50 and 79%? -Doc: It"s always the same! As soon as there"s HIV, it"s a rating from 50 to 79%.

That"s the problem: is that the equivalent of giving AAH?

-JPS: Why would we cancel it on him?

-Doc: Once we give 50-79, that makes it AAH! If we turn them down, they appeal it at the CDA [CDAPH] and they grant AAH.

-JPS: OK, we renew for how long? -Doc: (irritated) Ten years! What"s the point of asking the same thing every five years?

-JPS: But if the guidelines change… -Doc: In the CDA they want us to prove there"s been improvement. It works like an acquired right. The rating can"t be lowered below 50% even if there"s improvement in how the patient is doing.

-JPS: We"ll renew for five years.

Several relevant points can be drawn from this case by comparing it with the rest of our datanot because the incident is paradigmatic, but because its particularities shed light on how MDPH professionals use RQTH. We highlight three of them here:

1. It is rare for assistance or a form of recognition to be withdrawn at the time of a renewal request. Although the quoted example concerns AAH and a disability rating, this phenomenon is even more marked where RQTH is concerned. As it is, RQTH is rarely refused upon first request, and renewal is almost never declined. The existence of this "ratchet effect," as economists call it, arises from a conception of disability as a persistent limitation connected with permanent or even degenerative pathological processes, as opposed to a conception framed in terms of "situations of handicap" which would, for example, lead disability recognition to be revoked when a person suffering from an allergy is removed from the allergenic context.

2. In some cases, especially for AAH beneficiaries, interdisciplinary teams use RQTH status to signal that the applicant might in fact be able to find work, using it essentially as an injunction to work. The applicant above understood this very well, forcefully reacting in his letters. Our interviews with applicants indicate, however, that most of the time the signal does not come through very clearly.

3. One last contribution concerns the decision-making routine, rooted in a technicity emphasized by the proliferation of acronyms. The applicant"s personal situation, which may be described in a letter or the "life plan" section of the file, is often evoked at the beginning of the discussion but quickly forgotten by the time the decision is made, based on pre-established criteria in the great majority of cases. In this case, the doctor knows that he is defending a lost cause and tries to shake up the routine by suggesting, somewhat provocatively, some unusual possible decisions: not renewing AAH, imposing RQTH, reducing the disability rating, or in the other direction, renewing the AAH for ten years. But the decision ultimately conforms to expectations while taking account of the applicant"s refusal of RQTH.

More generally speaking, observation of the work carried out in MDPHs confirms that French disability policy remains fundamentally integrationist and has not taken the inclusive turn that the arrival of some concepts (such as those of the environment, accessibility, and situation) had nonetheless implied. The integrationist approach, clearly launched with the 1975 law, consists of compensating for disability with increased financial assistance and measures targeting persons recognized as disabled, and potentially their families as well. Policy for the employment of disabled persons is particularly representative of this movement, since it applies the principle of affirmative action. MDPHs prioritize the distribution of benefits according to the individual"s characteristics, not those of his or her situation. The emergence of the inclusive approach has had little impact on how rights to benefits are allocated. Moreover, full application of the situational logic would present serious problems for businesses under the current system: customizing a workstation would lead to the disappearance of the "situation of handicap," thus logically to the non-renewal of the employee"s RQTH, penalising the business for respecting the disabled worker quota.

We have seen that, contrary to the situational rationale, MDPHs work to make disability exist outside of "situations." The entire programme is based on the idea that the existence of "disability" is objective enough to allow the identification of disabled persons, disabled workers among them, or to put it another way, disabled persons whose limitations are not incompatible with exercising a job. And yet study of the applicants" perspective, which we turn to next, shows that the rationales of their claims are sometimes very different from those orienting MDPH decisions and in certain aspects may manifest a vision approaching that of situations of handicap.

Claims in their contexts

As Jean-François Ravaud, Isabelle Ville and Alain Letourmy have shown (2002), the issues behind self-declaring disability are complex. Although their article clearly identifies its connections with employment, or at least with "the economic value" of the "available individual potential" (Ibid: 549), many individuals nonetheless distinguish between work limitations and disability in general. In other terms, applicants can use the logic of the "situation of handicap" to set their cases apart from those of people considered to be "really" disabled. Especially in the field of disability, words are a medium for self-conception and self-presentation (even in the most bureaucratic language), and they give meaning to potentially painful events and processes. In France as in the United Kingdom [START_REF] Watson | Well, I Know this is Going to Sound Very Strange to You, but I Don't See Myself as a Disabled Person: Identity and disability[END_REF] and elsewhere, disability is still seen more as a stigma than as a positive component of selfidentity. This explains why our interviews with RQTH applicants revealed a considerably wider array of motives than our observations of the interdisciplinary teams and CDAPH had led us to expect.

"Work-related disability"

A non-negligible share of the applicants we met had very low expectations of their RQTH claims, which also means they were not very worried about them. For some, a claim has the sole value of being a prerequisite for access to a specialized setting: this is the case for Mrs Girard (a former computing specialist), who would like to enrol her daughter (who has a psychomotor developmental delay) in an experimental centre for mentally disabled youth; she filed a claim because she "saw on their website, it"s written that they take young people who are recognized as RQTH." Others made a claim simply because they were told to, but without really appropriating the process, as exemplified by Mrs Caillaud (a former house-cleaner) who had followed her social worker"s advice while specifying that she "wasn"t expecting anything special," or Mr Vanheck (a former tile-layer) who requested RQTH because his employer wanted it, but declared "I don"t know the advantages, I don"t know. I guess there are advantages, I have no idea." Others checked off multiple boxes on the application form, unable to distinguish clearly among them.

As for those who are more invested in the undertaking, many use a neutralization technique that makes a distinction between work-related disability and "real disability." Mrs Wilbrow, a former labourer of 45 years, explains it as follows: "I said to myself, I"m not going to make an application, I"m not disabled. But it"s a handicap for work, what they mean, it"s not disabled in a wheelchair, or…" And Mr Taget, a 56-year-old labourer, sees things the same way: "As for what I"ve got, I"m not disabled but, how to explain it… there are things I can"t do anymore." And he insists that his situation is different from that of one of his colleagues, who is "really disabled." With their word choice, these people use the notion of "situation of handicap": though they are bothered in certain situations (in this case, professional situations), this does not make them disabled persons.

Sometimes this desire for distance from "real" disability takes the form of worry, especially among parents seeking recognition for a child. Mrs Suguet (a public relations manager in a pharmaceutical company), who has a visually disabled 21-year-old daughter, clearly felt this way, declaring: "you have the impression that you"re no longer like everyone else, obviously, because you"re labelled, you find yourself in a category of disabled persons."

Consequently, there are many strategies for distancing a claim for RQTH from the identification of oneself or a loved one as a disabled person. In the process, many claimants emphasize the working conditions that reveal their limitations, which could be called circumstantial. In a way, it is their particular work situation that obliges them to appeal to the MDPH, and not their intrinsic limitations. But at the same time, their personal situations, in the more general sense of where they are in their life course, also explain the broader reasons behind their claims. he should be eligible for it, well then, he already has a foot in the door," Mrs Vermauld states. RQTH here is a way of progressively entering the field of disability and gradually getting comfortable with the idea that one could be thought of as disabled.

-A shield to let them stick it out. The second configuration, at the other extreme of the professional career, is that of older active workers worn down by work. In RQTH they hope to find protection from being laid off, a softening of their final years of professional activity, the reduction of heavy tasks, and/or a way to "hold on" for a few more years. In such cases people seem to see the RQTH as a sort of shield for sheltering themselves from difficult work. The RQTH claim of Mr Tajet, a labourer in a tense work setting, should be contextualized thusly. Proud of being a "daredevil" and not letting himself be pushed around, he is in conflict with his bosses on how work should be organized and has already received several warnings. He hopes that RQTH status will protect him from the risk of being let go and allow him to refuse certain tasks. In these situations, RQTH is rarely the manifestation of a self-perception as a disabled person and more often appears to be a strategic move against the boss and/or certain co-workers.

-The right not to work: The third configuration is composed of people with low occupational qualifications, often older and approaching retirement but sometimes younger, who are worn out by work, like the previous group, but who find themselves unemployed or with a very part-time job (a few hours of house cleaning, for example).

For them, RQTH is also a protection, but in this case turned against Pôle Emploi, the national job placement service. So it is that Mr Dahuet, a former agricultural labourer of 55 years of age suffering from phlebitis, asked for RQTH two years earlier to prove to Pôle Emploi (which he perceives as a hostile institution inattentive to people and their aspirations) that his health problems are not "imaginary." At his age, after nearly 40 years of labour, he does not see himself finding work again and would like to be left alone. In this case, RQTH allows resistance to social policies" logic of activation [START_REF] Barbier | Systems of social protection in Europe: Two contrasted paths to activation, and maybe a third[END_REF] by paradoxically reviving one of the primary functions of the notion of disability, allowing individuals to survive without working [START_REF] Ville | From inaptitude for work to trial of the self: The vicissitudes of meanings of disability[END_REF].

-The opportunity to change jobs: The fourth configuration includes people of all ages who are trying to re-integrate themselves into the job-market after having lost a job or taken a break from employment. Their RQTH claims, paired with a request for job training, are for them opportunities to change lines of work. Mrs Heermersch, a 52-year-old with 32 years as a labourer behind her, asked for RQTH after being laid off. Her goal was to take a course so she could change careers: "Well, I wanted to do itwhy? Because I seriously wanted to change the kind of work." The overlap of RQTH claim and life course change is maximal here: RQTH is taken almost as a justification, as a chance to restart one"s life and break away from one"s previous social trajectory [START_REF] Bidart | Time and Process: An Operational Framework for Processual Analysis[END_REF].

-An advantage for the business: The fifth and final configuration applies to cases where it is the employer rather than employee who presses for an application submission, in the primary interest of the business. Beyond the necessity of filling the quota of 6% disabled employees, businesses may use disability to their image, in that it is a factor in promoting "diversity" in the professional world [START_REF] Zanoni | Deconstructing difference: The rhetoric of human resource managers" diversity discourses[END_REF]. The example of Mr Leroy, manager in an architectural firm who has a congenital hearing disability in one ear, shows how such a strategic interest may be presented, in this case without great conviction, as a personal process: "I"ve been in the company for 25 years already, so there wasn"t any trouble. And then came a time… well, I knew that there was a disabled person in the company and I said to myself "Well, ok, when it comes down to it, am I disabled or not?"" Though it may be a stigma for individuals, disability can become an advantage for businesses. forward!"" And yet in the vast majority of cases, RQTH"s effects seem modest, thus disappointing for those who hoped for an improvement in their circumstances. The main expectations applicants expressed, presented above, are most often left frustrated: for one thing, the work protections seem rather weak, and the promise of professional integration rarely materializes. As a result, post-acquisition opinions of RQTH are generally ambivalent, and sometimes downright negative.

Conclusion

Employment policy favouring disabled persons, despite being foundational to the field of disability, has gradually been made into its own specific domain, which in turn undermines the unity of disability by producing disabled workers who often have trouble saying they are "really" disabled persons. And yet it is not the situational definition that used for recognizing disabled workers. French disability policy is consistent on this point, despite a few ambiguities: the integrationist rationale holds sway, fostering a conception of disabled persons as people with rights who should benefit from affirmative action bringing them closer to normalised social participation [START_REF] Winance | Being normally different? Changes to normalization processes: from alignment to work on the norm[END_REF]. Developments in their "situations" thus have little influence on the rights they might claim, since rights are still indexed on applicants" intrinsic characteristics, especially those of a medical nature.

Still, the notion of the "situation of handicap" does make sense from the applicants" perspective. They invoke it often, in other terms, to qualify their particular situations, between disability and normality. They, too, often consider themselves at disability"s fringes, affected by the limitations that they may relate to a variety of factors (age, physical wear and tear, illness, stress…) more or less independent of what disability means to them. From their

-

  SW: That didn"t work… (laughs) -JPS: The AFPA [the association for adult job training] saw him. No one has the file. He isn"t interested in training. -Doctor (male) [Doc]: He has a viral condition [a euphemism he frequently uses to say someone is infected with HIV] -Coord: Not only that, given the letters… -Doc: He"s an addict, seriously.

See the box "Methodology," below, for more detail.

The 2005 law increased this contribution, which is often seen as a penalty for not meeting the quota. Private businesses pay into one fund, and public employers into another.

This data comes from DARES (DARES Analyses, n° 70, November 2013).

This reform recommends suggesting to AAH-seekers that they have work capabilities that they would be able to put to use should they request and receive an RQTH instead. Some MDPHs follow this procedure and suggest RQTH to people who had not requested it, while others refuse to do so, believing that all claims for RQTH should come from the applicants themselves.

All names have been changed to respect the anonymity of the people concerned.

Not having recorded these meetings, we transcribed our notes here as precisely as possible.

The project-support service is charged with developing a list of professional options with the concerned person, following a series of tests, evaluations, and interviews. Applicants have access to the service on the recommendation of MDPH and may spend up to 40 hours with them over the course of six weeks.

The disability rating is more important in AAH attribution, for which applicants with an estimated rating below 50 % are not eligible. Applicants with a rating over 50 % but under 80 % (often shorthanded as "50-79") are eligible for the benefit only if the MDPH also acknowledges the existence of a "substantial and durable restriction of access to employment." The benefit is granted automatically when the disability rating is 80 % or higher.

discourse emerges the fact that recourse to the disability sector takes on very different meanings depending on the issues that concern them, which are connected in turn to their professional and biographical positions: integration into the job market, obtaining some protection from the violence of professional relations, changing careers, or gently winding down their working lives. Alongside these concerns, self-identification as a disabled person does not seem to be the most important or most emphasized issue.

A claim on protection taking many forms

Our interviews demonstrate the extent to which an RQTH claim may be partially or entirely disconnected from a claim for the recognition of a disability. Moreover, the quest for recognition is often secondary to a demand for one of many forms of social protection, as the following typology shows [START_REF] Bertrand | Devenir travailleur handicapé. Enjeux individuels, frontières institutionnelles[END_REF], allowing claim-related issues to be put into connection with the applicant"s life course and, more specifically, his or her position on the job market. Five configurations may thus be distinguished:

-Saying the unspeakable: The first configuration is that of young adults over the age of majority who are in the midst of transitioning into the job market. For them and their parents, there are two issues at stake in RQTH status: it is helpful for professional integration and it officially acknowledges the difficulties they face. When these difficulties are hard to see, especially in cases of a mental or psychological impairment, they are indeed likely to provoke misunderstanding and hostile behaviour from coworkers and hierarchical superiors. The situation of Mr Vermauld, age 23, allows us to illustrate. During the interview, his parents (a salesman and a specialised youth counsellor) explain that they have always tried to help their son be "drawn upward," especially by refusing to educate him in specialized institutions, but "it gets more complicated when it"s time to find a job." So they requested RQTH and fought so that he could be trained as an industrial cleaning agent. But RQTH also represents a way of expressing the "unspeakable" for them, to mitigate the fact that "people, they don"t understand." For these parents who until now have kept their distance from institutions for the disabled, this claim makes them feel like they have "crossed the line." They imagine the future a little more calmly now: "Recognitions for disabled workers today don"t give him access to an AAH, and we don"t want him to have AAH. But if someday

Generally disappointing responses

Can RQTH really answer all these expectations? The claims of most people we spoke with succeed. Although some expressed satisfaction, the predominant reaction is indifference tinged with disappointment or even disillusion. When Mrs Pottet (a 56-year-old labourer) obtained RQTH, it did not move her "one way or another," and she did not inform her employer. She knew that in a few months she would be leaving her company, which is in difficulty, and when a researcher asked her why she did not mention it, she answered: "What would that change?" Mrs Vanbingen, an unemployed woman of 44 years, was very disappointed when she initially got RQTH after having requested AAH. After an appeal she ultimately obtained AAH. The RQTH status is not worth very much to her; when asked how had been able to use it, she answers, actually referring to the AAH back-payment she had finally received, "I used it to buy a washing machine."

It should be said that there are few cases in our corpus where applicants experienced a significant change after receiving RQTH. We can cite Mr Odzinski"s case; a former labourer of 56 years who had been laid off, he got RQTH after having a stroke and found a part-time job in a high school cafeteria. Mr Karder is another example: a former laboratory assistant in his fifties suffering from silicosis, at Pôle Emploi he learned that local education authorities were looking for teachers with RQTH, and once he got it he was able to change careers quickly to become a middle school mathematics teacher. In addition, parents of young adults trying to enter the job market may see RQTH acquisition in a positive light because it represents a step forward in the process of employment integration. This is why Giacomo Mardo"s father awaited it with such impatience, to facilitate (or so he hoped) his son"s professional endeavours. Getting this recognition was a great comfort to him: "Yes, it reassured me, for sure. (…) I told myself, "well, ok, there"s one good thing, we can move


Saïd Rahmani
email: said.rahmani@emse.fr

Jean-Charles Pinoli
email: pinoli@emse.fr

Johan Debayle
email: debayle@emse.fr

Characterization and Estimation of the Variations of a Random Convex Set by its Mean n-Variogram : Application to the Boolean Model

Keywords: Boolean model, geometric covariogram, mixed volumes, particle size distribution, n points set probability, random set, shape variations

In this paper we propose a method to characterize and estimate the variations of a random convex set Ξ0 in terms of shape, size and direction. The mean n-variogram γ

] of a random convex set Ξ0 on R d reveals information on the n th order structure of Ξ0. Especially we will show that considering the mean n-variograms of the dilated random sets Ξ0 ⊕ rK by an homothetic convex family rKr>0, it's possible to estimate some characteristic of the n th order structure of Ξ0. If we make a judicious choice of K, it provides relevant measures of Ξ0 . Fortunately the germ-grain model is stable by convex dilatations, furthermore the mean n-variogram of the primary grain is estimable in several type of stationary germ-grain models by the so called n-points probability function. Here we will only focus on the Boolean model, in the planar case we will show how to estimate the n th order structure of the random vector composed by the mixed volumes t (A(Ξ0), W (Ξ0, K)) of the primary grain, and we will describe a procedure to do it from a realization of the Boolean model in a bounded window. We will prove that this knowledge for all convex body K is sufficient to fully characterize the so called difference body of the grain Ξ0 ⊕ Ξ0. we will be discussing the choice of the element K, by choosing a ball, the mixed volumes coincide with the Minkowski's functional of Ξ0 therefore we obtain the moments of the random vector composed of the area and perimeter t (A(Ξ0), U (Ξ)) . By choosing a segment oriented by θ we obtain estimates for the moments of the random vector composed by the area and the Ferret's diameter in the direction θ, t ((A(Ξ0), HΞ 0 (θ)). Finally, we will evaluate the performance of the method on a Boolean model with rectangular grain for the estimation of the second order moments of the random vectors t (A(Ξ0), U (Ξ0)) and t ((A(Ξ0), HΞ 0 (θ)).

Introduction

A random closed set (RACS) denotes a random variable defined on a probability space (Ω, A, P ) taking values in (F, F) the family of all closed subset of R d provided with the σ-algebra F = σ{{F ∈ F |F ∩ X = ∅} X ∈ K} where K denotes the class of compact subsets on R d . As when we work with random vectors it is necessary to give meaning to the concept of distribution. Choquet and Matheron have shown that a random set is fully characterized by its probability of presence in each place of the space, thus the concept of distribution is replaced by the so called functional capacity also called Choquet capacity

T Ξ : K → [0, 1] . T Ξ (X) = P ({Ξ ∩ X = ∅}) (1)
Several materials can be modeled by random sets. In fact, the heterogeneity of the materials can be apprehended by a probabilistic approach [START_REF] Torquato | Random heterogeneous materials: microstructure and macroscopic properties[END_REF], [START_REF] Jeulin | Random texture models for material structures[END_REF]. Especially granular or fibrous media [START_REF] Galerne | Modèles d'image aléatoires et synthèse de texture[END_REF], [START_REF] Peyrega | Prediction des proprietes acoustiques de materiaux fibreux heterogenes a partir de leur microstructure 3D[END_REF] can be represented by unions of overlapping particles (the grains) centred on random positions (the germs), thus giving rise to the germ-grain model.

Ξ = xi∈Φ Ξ i + x i ( 2
)
Where Φ is point process (REF) which generates the germs x i , and the grains Ξ i are convex random sets independent and identically distributed. Notice that this definition assumes the independence between the particles Ξ i and their positions x i , there is a more general definition authorizing the correlation between germs and grains [START_REF] Chiu | Stochastic geometry and its applications[END_REF], for more convenience we choose to introduce the model under this hypothesis. There are two types of use of this model. The first one is the simulation of a material, the global characteristics of the model match to the material's ones but the germs and grains have no physical sense: the local characteristics of the model (Φ and Ξ 0 ) a priori have no connection with the intrinsic structure of the material. Our approach consist in representing the people of crystals by such a model; that is to say that the point process Φ is the repartition of the particles and the convex random sets Ξ i the particles themselves. The goal is to adjust the model to actual data from measurements acquired by an image acquisition system. Generally, the acquisition is obtained by optical imaging, in other words, we have realizations of Ξ ∩ W where W is a bounded window and we want to estimate the characteristics of Φ and Ξ 0 . To meet this objective, we focuses on two points: first, estimate the characteristics of Ξ from a realization of Ξ ∩ W [START_REF] Molchanov | Statistics of the Boolean model for practitioners and mathematicians[END_REF], secondly establish relationships between characteristics of Ξ and the local characteristics of the model (Φ, Ξ 0 ). This second point raises the problem of non-uniqueness of the representation (2). To remedy this, we introduce an additional assumption: we assume the process Φ comes from a known type (Poisson process, Cox process, ...). We will use the homogeneous Boolean model, a germ-grain model in which Φ is a homogeneous Poisson point process. This model is widely used because we have an analytical formula for the Choquet capacity.

T Ξ (X) = 1 -exp(-λE[ν d (Ξ 0 ⊕ X)]) , ∀X ∈ K (3)
Several methods are used to connect the global characteristics of the model to the characteristics of the primary grain. In the plane and the space, Miles's formulae [START_REF] Miles | Estimating aggregate and overall characteristics from thick sections by transmission microscopy[END_REF] or minimum contrast method [START_REF] Stoyan | Stochastic Geometry and its application[END_REF] can estimate the average value of Minkowski's functional of the primary grain. Generally, the primary grain is assumed to have a known and deterministic shape, that is to say, the realizations of the primary grain are homothetic. So as to estimate the variations of the scaling factor from the expectation of the Minkowski functionals. For example, for a disc in the plane, the moments of the first and second orders of the radius of the primary grain are respectively proportional to its average perimeter and its average area. However, if we consider that the shape of the grain can vary, several issues remain unresolved: firstly Minkowski functionals of a random convex set are not enough to characterize its shape and also their average will not provide a sufficient information to characterize its variations. For instance, for a Boolean model whose grain has a shape that depends on several parameters (rectangle, ellipse ...), the estimation of geometric variations of the grain is not direct. The aim of our work is to characterize and estimate the variations of the primary grain of the Boolean model without any assumption concerning its shape, from a realization of the model in a bounded window. The ideal would be to estimate the functional capacity of the primary grain T Ξ0 or an equivalent which completely characterizes the random convex Ξ 0 . Applying the Steiner's formula and the linearity of the expectation, for all convex compact set X we have:

E[ν d (Ξ 0 ⊕ X)] = d k=0 d k E[W d-k (Ξ 0 , X)] (4)
Where W d-k (Ξ 0 , X) denotes the (d -k) th mixed volume of Ξ 0 and X. Considering the relationship (3), we understand that the functional capacity of the model evaluated on a convex K depends on the grain only by the expectations of mixed volumes between Ξ 0 and X. In order to reveal the variations of Ξ 0 we need to consider the functional capacity of the model on compacts that are not convex, that is why we are interested in the n-pointprobability function [START_REF] Chiu | Stochastic geometry and its applications[END_REF]:

P({x 1 , • • • x n } ⊂ Ξ).
For n = 2, this quantity is known under the name of covariance and it can be connected to the mean covariogram of the primary grain [START_REF] Galerne | Computation of the perimeter of measurable sets via their covariogram. applications to random sets[END_REF]. For any n ≥ 3, the n-point-probability function can be used to estimate the mean n-variogram γ

(n) Ξ0 : (u 1 • • • u n ) → E[ν d (Ξ 0 ∩ (Ξ 0 -u 1 ) • • • ∩ (Ξ 0 -u n ))
] of Ξ 0 , this quantity evaluated on the dilated grain Ξ 0 ⊕ K for a convex compact set K reveals the n th order structure of Ξ 0 , especially some linear combinations of the expectations E[

d k=0 W d,k (Ξ 0 , K) p k ] of order p = d k=0 p k ≤ n where W d,k (Ξ, K) , k = 0, • • • d,
denotes the mixed volumes of Ξ 0 by K. First we will discuss the properties of the mean n-variogram of a random convex and how it describes its n th order structure. Secondly we will focus on the planar case, we will discuss the interpretation of the n th order moments of the vector t (A(Ξ 0 ), W (Ξ, K)), and we will show how they can be estimated for the primary grain of a boolean model. Finally we will test the estimation method by the simulations of a boolean model with rectangular grains.

From the Mean n-Variogram of a Random Convex Set to its Variations

In this section we will discuss the properties of the the mean n-variogram of a random convex and how it describes his n th order structure.

Mean n-Variogram of Random Convex Set

In this paragraph we will define the mean n-variogram of a random convex and discuss its properties. The mean n-variogram is a simple generalization of the concept of mean covariogram introduced by Bruno Galerne in [START_REF] Galerne | Computation of the perimeter of measurable sets via their covariogram. applications to random sets[END_REF] for n = 2. The proof of the following results can be easy found by recursion on n from Bruno Galerne's proof [START_REF] Galerne | Computation of the perimeter of measurable sets via their covariogram. applications to random sets[END_REF], that is why we will omit them.

Definition 1. Let Ξ 0 be a random convex set satisfying E[ν d (Ξ 0 )] < ∞ and n ≥ 1 we will call mean n-variogram of Ξ 0 the expectation of its n-variogram : γ (n) Ξ0 : R d×(n-1) -→ R + (u 1 , • • • u n-1 ) -→ E[ν d ( n-1 i=1 (Ξ 0 -u i ) ∩ Ξ 0 )] Proposition 1. Let Ξ 0 be a random convex set satisfying E[A(Ξ 0 ) n ]
< ∞ and n ≥ 3 (the cases n < 3 were treated in [START_REF] Galerne | Computation of the perimeter of measurable sets via their covariogram. applications to random sets[END_REF]).Then the mean n-variogram have the following properties:

i) permutation invariant: ∀σ ∈ S n , γ (n) Ξ0 (u 1 , • • • u n-1 ) = γ (n) Ξ0 (u σ(1) , • • • u σ(n-1) ) ii) Reducibility: ∃i = j, u i = u j ⇒ γ (n) Ξ0 (u 1 , • • • u n-1 ) = γ (n-1) Ξ0 (u 1 , • • • u i-1 , u i+1 , • • • u n-1 )
and ∃i,

u i = 0 ⇒ γ (n) Ξ0 (u 1 , • • • u n-1 ) = γ (n-1) Ξ0 (u 1 , • • • u i-1 , u i+1 , • • • u n-1 ) iii) ∀(u 1 , • • • u n-1 ) ∈ R d×(n-1) , 0 ≤ γ (n-1) Ξ0 (u 1 , • • • u n-1 ) ≤ γ (n-1) Ξ0 (u 2 , • • • u n-1 ) iv) ∀k ≤ n -1, R d • • • R d γ (n) Ξ0 (u 1 , • • • u n-1 )du 1 • • • du k = ... E[ν d (Ξ 0 ) k ]γ (n-k) Ξ0 (u k+1 , • • • u n-1 )
especially for k = n we have:

R d • • • R d γ (n) Ξ0 (u 1 , • • • u n-1 )du 1 • • • du n-1 = E[ν d (Ξ 0 ) n ] (5) v) ∀(u 1 , • • • u n-1 ) ∈ R d×(n-1) , γ (n) Ξ0 (-u 1 , • • • -u n ) = γ (n) -Ξ0 (u 1 , • • • u n ) and γ (n) Ξ0 (-u 1 , u 2 • • • u n-1 ) = γ (n) Ξ0 (u 1 , u 2 -u 1 , • • • u n-1 -u 1 ). vi) The partial map u → γ (n) Ξ0 (u 1 , • • • u n-2 , u
) is uniformly continuous and zero limit when u → +∞. Furthermore for all u ∈ R d the map r → γ

(n) Ξ0 (u 1 , • • • u n-2 , ru) is decreasing on R. vii) γ (n)
Ξ0 also has an integral formulation:

γ (n) Ξ0 (u 1 , • • • u n-1 ) = R d P({x, x + u 1 , • • • x + u n-1 } ⊂ Ξ 0 )dx (6)
The relationship (5) has a the great advantage of giving access to the n th order moment of the volume of the random set Ξ 0 . This relationship is even more important because dilating Ξ 0 by a convex K, it gives access to some linear combinations of the expectations E[

d k=0 W d,k (Ξ 0 , K) p k ] of order p = d k=0 p k ≤ n .

Dilatation of a Random Convex Set

Let Ξ 0 be a convex random set and K a convex compact set, let's recall the Steiner's formula:

∀r ≥ 0, ν d (Ξ 0 ⊕ rK) = d k=0 d k W d,k (Ξ 0 , K)r k (7)
Where W d,k (Ξ 0 , K) denote the mixed volume of k-homogeneity in its first variable (d-k)-homogeneity in its second, that is:

∀k = 0, • • • d, ∀(α, β) ∈ R 2 + , W d,k (αΞ 0 , βK) = α d-k β k W d,k (Ξ 0 , K) (8) Furthermore W d,d (Ξ, K) = ν d (K) and W d,0 (Ξ, K) = ν d (Ξ). For r ∈ R + , n ≥ 0 and Ξ 0 satisfiying E[ν d (Ξ 0 ) n ] < ∞
we introduce the function:

ζ (n) Ξ0,K : R + -→ R + r -→ E[ν d (Ξ 0 ⊕ rK) n ] (9)
The existance of ζ

(n) Ξ0,K (r) = R d • • • R d γ (n) Ξ⊕rK (u 1 , • • • u n-1 )du 1 • • • du n-1 (10)
by injecting [START_REF] Miles | Estimating aggregate and overall characteristics from thick sections by transmission microscopy[END_REF] in [START_REF] Galerne | Computation of the perimeter of measurable sets via their covariogram. applications to random sets[END_REF] we have:

ζ (n) Ξ0,K (r) = E[( d k=0 d k W d,k (Ξ 0 , K)r k ) n ] (11) ζ (n)
Ξ0,K is therefore a polynomial function in r of degree n × d it can be expressed as:

ζ (n) Ξ0,K = nd j=0 C (K) n,j r j (12)
using the multinomial theorem, each of these coefficients

C (K)
n,j of degree j ≤ nd, can be expressed as a linear combination of the interactions E[ Ξ0,K in a similar way to the minimum contrast method [START_REF] Heinrich | Asymptotic properties of minimum contrast estimators for parameters of boolean models[END_REF]. Unfortunately in the general case it is not sufficient to obtain the interactions

d k=0 W d,k (Ξ 0 , K) p k ] satisfying j =
E[ d k=0 W d,k (Ξ 0 , K) p k ] from the coefficients of ζ (n)
Ξ0,K . However in the planar case, we can do this as follows. Theorem 1. Let Ξ 0 be a convex random set on the plane R 2 , we introduce the polynomial function η (n) Ξ0,K as follows:

η (n) Ξ0,K (r) = n j=0 n j (-1) j A(K) j r 2j ζ (n-j) Ξ0,K (r) (13)
Where A(K) = ν 2 (K) denotes the area of K. Then η

(n)
Ξ0,K is a polynomial function of degree n and if we note

M (K) n,k
its k th order coefficient for all n ∈ N and k = 1, • • • n, we have:

M (K) n,k = k 2 j=0 (-1) j A(K) j n j C (K) 2n-2j,k-2j (14)
and

E[A(Ξ) n-k W (Ξ 0 , K) k ] = M (K) n,k 2 k n k ( 15
)
Where W (Ξ 0 , K) = W 2,1 (K) denotes the mixed area and k 2 denotes the floor of k 2 . Proof. First according to Steiner's formula,

A(Ξ 0 ) + 2rW (Ξ 0 , K) = A(Ξ 0 ⊕ rK) -r 2 A(K) ⇒ E[(A(Ξ 0 ) + 2rW (Ξ 0 , K)) n ] = E[(A(Ξ 0 ⊕ K) -r 2 A(K)) n ]
by applying the binomial theoerm on each side of the equality and according to linearity of the expectation we have:

η (n) Ξ0,K (r) = n k=0 2 k n k r k E[A(Ξ) n-k W (Ξ 0 , K) k ] (16)
it follows the relationship [START_REF] Yau | Institute for Advanced Study[END_REF]. Injecting [START_REF] Schneider | Convex bodies: the Brunn-Minkowski theory[END_REF] in [START_REF] Stoyanov | Krein condition[END_REF] we have:

η (n) Ξ0,K (r) = n j=0 2(n-j) p=0 n j (-1) j A(K) j r 2j+p C (K) 2n-2j,p
applying the change of variable z = 2j + p we have:

η (n) Ξ0,K (r) = n j=0 2n z=2j n j (-1) j A(K) j r z C (K) 2n-2j,z-2j = 2n z=0 { z 2 j=0 n j (-1) j A(K) j C (K)
2n-2j,z-2j }r z by identification with ( 16) it follows the relationship ( 14).

Remark 1. The Theorem 1 shows how the n th order interactions

E[A(Ξ) n-k W (Ξ 0 , K) k ], k = 0, • • • n, can be estimated by the knowledge of the functions ζ (j) Ξ0,K , j = 1, • • • n.
We emphasize that for all convex compact K the distribution of the random vector t (A(Ξ), W (Ξ 0 , K)) is fully characterized and can be reconstructed from its moments: the interactions

E[A(Ξ) n-k W (Ξ 0 , K) k ], n ∈ N, k = 1, • • • n.
There is two way to estimate the n th order interactions: make polynomials approximations of the ζ (j) Ξ0,K , j = 1, • • • n to get their coefficients C (K) j,p and using [START_REF] Akhiezer | The classical moment problem: and some related questions in analysis[END_REF] or make directly a polynomial approximation of η (n) Ξ0,K to get the coefficients M (K) n,k and using (15).

Variation of a Random Convex Set in R 2

Here we will discuss the choice of K, let Ξ 0 be a convex random set of R 2 satisfying E[A(Ξ 0 )] < ∞. Let's note B the unit ball of R 2 and S θ the rotation of the segment [0, 1] × {0} with angle θ ∈ [0, 2π]; in other words S θ is the centred segment directed by θ of length two. We have the well know result [START_REF] Michielsen | Integral-geometry morphological image analysis[END_REF], [START_REF] Molchanov | Statistics of the Boolean model for practitioners and mathematicians[END_REF]:

W (Ξ 0 , B) = 1 2 U (Ξ 0 ) (17)
W (Ξ 0 , S θ ) = H Ξ0 (θ) (18)
Where U (Ξ 0 ) denotes the perimeter of Ξ 0 and H Ξ0 (θ) denotes the Ferret's diameter of Ξ 0 in the direction θ. As a direct result, the choice K = 2B provides estimators for all moments of the random vector t (A(Ξ 0 ), U (Ξ 0 )), and by choosing K = S θ we obtain all moments of the random vector t (A(Ξ 0 ), H Ξ0 (θ)). Notice that the Ferret's diameter is π-periodic in the variable θ, if Ξ 0 is supposed to be isotropic, then the random variables H Ξ0 (θ) for θ ∈ [0, π] are identically distributed. Let's remark that the random process H Ξ0 = (H Ξ0 (θ)) θ∈[0,π] fully characterize the random set Ξ 0 ⊕ Ξ0 ; in fact, for each ω ∈ Ω the Ferret's diameter H Ξ0 (ω) : [0, π] → R + coincide with the support function of the convex compact set Ξ 0 ⊕ Ξ0 (ω). It is well known that the support function of the convex compact set fully characterizes the convex compact set concerned [START_REF] Schneider | Convex bodies: the Brunn-Minkowski theory[END_REF]. Therefore H Ξ0 fully characterizes the difference body Ξ 0 ⊕ Ξ0 that leads to the following theorem.

Theorem 2 (Characterization of a random convex set by its mixed area). Let's Ξ

(1) 0 and Ξ

(2) 0 be two convex random sets of R 2 satisfying E[A(Ξ (j) )] < ∞, j = 1, 2 and assume at least one of the distributions of the random variables A(Ξ (j) ) is M-determinate [START_REF] Stoyanov | Krein condition[END_REF]. Then, the condition

∀n ≥ 1, ∀K ∈ K c , ζ (n) Ξ (1) 0 ,K = ζ (n) Ξ (2) 0 ,K (19) Implies, Ξ (1)
0 ⊕ Ξ(1) 0 L = Ξ (2) 0 ⊕ Ξ(2) 0 ( 20
)
Where " L =" denotes the so called equality in law, and K c denotes the class of convex compact sets on R 2 .

Proof. Let's Ξ

(1) 0 and Ξ

(2) 0 be two convex random sets of R 2 satisfying E[A(Ξ (j) 0 )] < ∞, j = 1, 2 and assume at least one of the distributions of the random variables A(Ξ (j) ) is M-determinate Let's assume the condition (19), according to the Theorem 1 and the M-determinate condition:

(19) ⇒ ∀n ≥ 1, ∀K ∈ K, E[W (Ξ (1) 0 , K) n ] = [W (Ξ (2) 0 , K) n ] ⇒ W (Ξ (1) 0 , K) L = W (Ξ (2) 0 , K) For each k ≥ 1 and for each (θ 1 , • • • θ k ) ∈ [0, π] k , let V 1 = t (H Ξ (1) 0 (θ 1 ), • • • H Ξ (1) 0 (θ k )) be a random vector extract of the random process H Ξ1 and V 2 = t (H Ξ (2) 0 (θ 1 ), • • • H Ξ (2) 0 (θ k )) a random vector extract of H Ξ (2) 0 . We will prove V 1 L = V 2 , for this, let's consider a positive linear combination of elements of V 1 , k i=1 α i H Ξ
W (X, Y ⊕ xS β ) = W (X, Y ) + xH X (β) (21)
Applaying successively this lemma on W (Ξ

0 , Z) and on W (Ξ

0 , Z) we have:

W (Ξ (1)
0 , Z) = k i=1 α i H Ξ (1) 0 (θ i ) and W (Ξ (2) 0 , Z) = k i=1 α i H Ξ (2) 0 (θ i ) Thus, ∀(α 1 , • • • α k ) ∈ R k + , k i=1 α i H Ξ (1) 0 (θ i ) = k i=1 α i H Ξ (2) 0 (θ i ) which implies V 1 L = V 2 ,
we have this result for all k ≥ 1 and for all (θ 1 ,

• • • θ k ) ∈ [0, π] k , thus H Ξ (1) 0 L = H Ξ (2) 0 therefore Ξ (1) 0 ⊕ Ξ(1) 0 L = Ξ (2) 0 ⊕ Ξ(2) 0 . Remark 2.
First note that the M-determinate condition is not realy restrictive, it can always be assumed in pratical cases [START_REF] Akhiezer | The classical moment problem: and some related questions in analysis[END_REF]. Let's notice that the choice of the convex compact Z = k i=1 α i S θi provide an explicit expression of the mixed area, it can be use for estimate all characteristics of the random process H Ξ0 in the following way; For each k ≥ 1 and n ≥ 0 we define

P (k,n) Ξ0,(θ1,•••θ k )
, the polynomial function on k variables of degree n as:

P (k,n) Ξ0,(θ1,•••θ k ) (α 1 , • • • α k ) =E[W (Ξ 0 , k i=1 α i S θi ) n ] =( k i=1 α i H Ξ0 (θ i )) n = j1+•••j k =n n j 1 , • • • j k E[ k i=1 H Ξ0 (θ i ) ji ] k i=1 α ji i Thus all of the expectations E[ k i=1 H Ξ0 (θ i ) ji ] can be estimate by a fit of P (k,n) Ξ0,(θ1,•••θ k ) .
However, in practice it is difficult to compute and fit P (k,n) Ξ0,(θ1,•••θ k ) for large k, thus we will be more interested in the autocorrelation of the random process H Ξ0 and its marginals moments E[H Ξ0 (θ) n ].

Application to the Boolean Model

Let Ξ be a Boolean of primary grain Ξ 0 and intensity λ :

Ξ = xi∈Φ Ξ i + x i ( 22
)
Where Φ is a homogeneous Poisson point process of intensity λ and the Ξ i are random convex sets identically distributed as Ξ 0 .

The Polynomial ζ (K)

Ξ0 for the Primary Grain of the Boolean Model Let's recall the fundamental relationship of functional capacity:

∀X ∈ K, T Ξ (X) =P ({Ξ ∩ X = ∅}) =1 -exp(-λE[ν d (Ξ 0 ⊕ X)]) =1 -exp(-Ψ Ξ (X)) Where Ψ Ξ (X) = -ln(1 -T Ξ (X)) = λE[ν d (Ξ 0 ⊕ X)].
Let's enunciate a useful lemma the proof is omitted since it can be established by induction.

Lemma 2 (inclusion-exclusion principle). Let f be a C-additive sets function, and (A i ) 1≤i≤n be a non degenerate family of subset of R n , then:

f ( n i=1 A i ) = n k=1 (-1) k+1 k! (i1,•••i k )∈I k f ( k j=1 A ij ) f ( n i=1 A i ) = n k=1 (-1) k+1 k! (i1,•••i k )∈I k f ( k j=1 A ij )
Where

I k = {(i 1 , • • • i k ) ∈ {i 1 , • • • i k } ⊂ {1, • • • n} k | ∀l ≤ k, ∀m ≤ k, l = m ⇒ i l = i m }.
For n ≥ 2 let's note u n = 0, according the Lemma 2 and the expression of Ψ , the mean n-variogram can be expressed as:

γ (n) Ξ0 (u 1 , • • • u n-1 ) =E[ν d ( n i=1 Ξ 0 -u i )] = n k=1 (-1) k+1 k! (i(1),•••i(k))∈I k E[ν d ( k j=1 Ξ 0 -u i(j) )] = n k=1 (-1) k+1 λk! (i(1),•••i(k))∈I k Ψ Ξ ({u i(1) , • • • u i(k) }) ⇒ γ (n) Ξ0 (u 1 , • • • u n-1 ) = n k=1 (-1) k λk! (i(1),•••i(k))∈I k ln(1 -T Ξ ({u i(1) , • • • u i(k) })) (23)
Obviously the quantities Ψ Ξ ({u i1 , • • • u i k }) can be expressed by the n-point probability function

C (n) Ξ (x 1 , • • • x n ) = P({x 1 , • • • x n } ⊂ Ξ) evaluated on the subsets of {u 1 , • • • u n }, we have: ln(1 -T Ξ ({u i1 , • • • u i k })) = ln(P( k j=1 {u ij / ∈ Ξ})) = ln( k j=1 (-1) j+1 j! (z1,•••zj )∈Ij P( j l=1 {u i(z l ) / ∈ Ξ})) = ln( k j=1 (-1) j+1 j! (z1,•••zj )∈Ij (1 -C (j) Ξ (u i(z1) , • • • u i(zj ) )) ⇒ γ (n) Ξ0 (u 1 , • • • u n-1 ) = n k=1 (i(1),•••i(k))∈I k (-1) k λk! ln( k j=1 (-1) j+1 j! (z1,•••zj )∈Ij (1 -C (j) Ξ (u i(z1) , • • • u i(zj ) )) (24)
The n-point-probability function C

(n)

Ξ (x 1 , • • • x n ) can be viewed as a volume fraction of n i=1 (Ξ + x i ), it is easy to see that the stationary of Ξ implies P({x 1 , • • • x n } ⊂ Ξ) = P({0, x 1 -x n , • • • x n-1 -x n } ⊂ Ξ)
,its yield an unbiased estimator for the n-point probability function in bounded windows W :

Ĉ (n) Ξ,W (x 1 , • • • x n ) = ν d ((Ξ ∩ W ) {0, x 1 -x n , • • • x n-1 -x n }) ν d (W {0, x 1 -x n , • • • x n-1 -x n }) (25)
Thus, an estimator γ(n) Ξ0,W for the mean n-variogram of Ξ 0 can be obtained by ( 25),(24) associated to an estimator of λ (see [START_REF] Miles | Estimating aggregate and overall characteristics from thick sections by transmission microscopy[END_REF], [START_REF] Chiu | Stochastic geometry and its applications[END_REF]). We emphasize that the Boolean model is stable by convex dilatation, in other words, for K ∈ K c , the dilated model Ξ ⊕K is also a Boolean model of same intensity λ and of primary grain Ξ 0 ⊕K. As a consequence, for each r ≥ 0 an estimator γ(n) Ξ0⊕rK can be found. However a precaution must be taken to break the edge effects; if we have a realization of Ξ ∩ W , the dilated model Ξ ⊕ rK is only known within the eroded window W rK = W rK, in fact ((Ξ ∩ W ) ⊕ rK) ∩ W rK = (Ξ ⊕ rK) ∩ W rK . Therefore it follow from (10) the estimators:

ζ(n) Ξ0,K,W (r) = R d • • • R d γ(n) Ξ⊕rK,W rK (u 1 , • • • u n-1 )du 1 • • • du n-1 (26)

The Case of the Planar Boolean Model

As a consequence of (26) and the Theorem 1, in the planar case we obtain an estimator for the polynomial η

(n) Ξ,K : η(n) Ξ0,K,W (r) = n j=0 n j (-1) j A(K) j r 2j ζ(n-j) Ξ0,K,W (r) (27)
Therefore by fitting these quantities, we obtain estimators for the moments of t (A(Ξ 0 ), W (Ξ 0 , K)). Furthermore the polynomial approximation of η(n) Ξ0,K,W can be refined by inequality constraints, some of them are probabilistic( inequality between moments, Cauchy-Schwarz inequality). But there are also some morphological constraints like the generalized isoperimetric inequality [START_REF] Yau | Institute for Advanced Study[END_REF]. Furthermore, if we make additive assumptions concerning the shape of Ξ 0 , other constraints can be found [START_REF] Rivollier | Analyse morphométrique d'images à tons de gris par diagrammes de forme[END_REF]. We have tested this method with n = 2 for a Boolean model of rectangular grains: the uncorrelated sides of the primary grain follow N (40, 10) and N (30, 10), the intensity parameter λ = 100 500×500 (see Figure 1). We have simulated several realization of this model in a bounded window W = 500 × 500, and we studies the relative error between the theoretical and estimated moments of t (A(Ξ 0 ), W (Ξ 0 , K)) when K is a ball or a segment (see Figure 1).

Conclusions and Prospects

We have established analytical formulae which allow us to connect the n-variograms of the dilatations of a random convex set with the variations of some of its morphological characteristics. For the Boolean model, we also have shown how the n-variogram of its primary grain can be connected to its n-point probability function. Therefore, it provides estimators of the variations of the primary grain's morphological characteristics. Especially, using dilatation by a disk or a segment, the proposed method can be used to characterize a primary grain whose shape depends on two parameters (rectangle, ellipse,...). We emphasize that our method can be used for any germ-grain model in which we can estimate the mean n-variogram.

In the future we are looking at more complex germ-grain models than the Boolean model. We are also interested in the influence of the model parameters and the observation's window on the accuracy of the estimates. The prospect of describing a convex random set by the characteristics of the random process associated to its Ferret's diameter (see subsection 2.3), is even more relevant.

  (n) Ξ0,K is ensured by the existance of E[ν d (Ξ 0 ) n ] and the convexity of Ξ 0 . Notice that for n ≥ 2 the functional ζ (n) Ξ0,K can be connected to the mean n-variogram of Ξ 0 ⊕ rK by (5) we have:

  ζ

d

  k=0 k × p k and d k=0 p k = n. Of course we can extract all coefficients C (K) n,j by searching a polynomial approximation of ζ (n)

  ) and the convex compact sets Z = k i=1 α i S θi . Notice the following property: Lemma 1. Let's X, Y be convex sets, x ∈ R + and β ∈ [0, π]. Using the Steiner's formula on A(X ⊕ Y ⊕ xS β ) and the properties of the support function, it is easy to show that:

Fig. 1 :

 1 Fig. 1: A realization of the test model (a) and the relatives error of the estimation of the 2 nd order moments of t (A(Ξ 0 ), U (Ξ 0 )) on (b) and of t (A(Ξ 0 ), H Ξ0 (θ)) on (c).