Dr Federico Ferretti -Ba
email: federico.ferretti@ucd.ie

Marie-Louise Berneri

Marie- Louise Berneri

Savonarola ' Camillo Berneri

Journeying through Utopia: anarchism, geographical imagination and performative futures in

Keywords: Berneri, geographical imagination, anarchism, utopianism, anti-colonialism Palabras llave: Berneri, imaginación geográfica, anarquismo, utopismo, anticolonialismo

, author of a neglected but very insightful history of utopias and of their spaces. Extending current literature on anarchist geographies, utopianism and on the relation between geography and the humanities, I argue that a distinction between authoritarian and libertarian utopias is key to understanding the political relevance of the notion of utopia, which is also a matter of space and geographical imagination. Berneri's criticisms to utopia were eventually informed by notions of anti-colonialism and anti-authoritarianism, especially referred to her original critique of twentieth-century totalitarian regimes. Then, I argue for a connection between anarchist, humanistic, cultural and historical approaches to geography, to extend the empirical and theoretical reach of the discipline and its relations with the 'humanities'. This paper likewise contributes to recent scholarship on transnational anarchism, arguing that the anarchist tradition cannot be understood outside its transnational, cosmopolite and multilingual networks and concrete practices: therefore, only relational, contextual and space-sensitive approaches can make sense of its specificity.

Resumen: Este articulo encara los trabajos y archivos de la militante anarquista transnacional Maria Luisa Berneri (1918Berneri ( -1949)), autora de un estudio poco conocido pero muy significativo sobre las historias de las utopías y sus espacios. Contribuyendo para literatura actual sobre geografías anarquistas, utopismo y sobre la relación entre la geografía y las 'humanidades', afirmo que una distinción entre utopías libertarias y utopías autoritarias es esencial para comprender la importancia política del concepto de utopía, que es también un asunto de espacio y de imaginación geográfica. Las críticas de Berneri a la utopía se inspiraron de su anticolonialismo y antiautoritarismo, focalizado especialmente en su original critica de los regímenes totalitarios del siglo veinte. Enseguida, propongo una conexión entre abordajes anarquistas, humanistas, culturales e históricos de la geografía para ampliar el campo empírico y teórico de la disciplina y de sus relaciones con las humanidades. Este articulo contribuye también a las investigaciones recientes sobre el anarquismo transnacional, afirmando que la tradición anarquista no puede ser comprendida fuera de sus redes transnacionales, multilingües y cosmopolitas y de sus practicas concretas. Entonces, solo lecturas contextuales, relacionales y sensibles al espacio pueden dar censo a la especificidad del anarquismo.

Introduction 1

This paper discusses the problem of how to build geographies of Utopia, drawing upon an exceptional and little-known case, that is the works of anarchist transnational intellectual and activist Maria Luisa (or Marie-Louise) Berneri (1918Berneri ( -1949)), author of neglected but exceptional works on histories and spaces of utopias. While her masterpiece Journey Through Utopia [START_REF] Berneri | Neither east nor west: selected writings[END_REF], written in the 1940s, addressed philosophical matters from the Antiquity to the beginning of the twentieth century, it was fully inserted in the context of the author's political activities.

There, first of all was the challenge of providing a suitable alternative to the dominant representations of the opposition between democracies and totalitarian regimes, a very urgent task for the international anarchist movement at that time. Berneri's reflections provide useful conceptual tools to understand anarchist approaches to the notion of utopia, and at the same time it contributes to current reflections on the relation between geography and utopia, extending scholarship on cultural, (neo)humanistic and anarchist geographies.

One of the political movements which have been especially identified with the idea of utopia is anarchism. This association is exemplified by the opening quotation from Camillo Berneri (1897Berneri ( -1937)), important transnational activist of the first half of the twentieth century and father of Maria Luisa, exposing a certain ambivalence of the notion of utopia in the anarchist intellectual tradition.

On the one hand, anarchists have often expressed appraisals of utopias as positive 'dreams' of a new society. On the other, they generally claim not to be utopians but proponents of real alternatives to the present society, arguing that 'utopia' is a bourgeois notion: Camillo Berneri's classification of the 'utopian' among the 'politicians' is a good example of this critique.

Anarchism is an idea which appears especially relevant for geography, given the historical links between the construction of an anarchist thinking and a geographical one embodied by figures such as Elisée Reclus (1830Reclus ( -1905) ) and Pyotr Kropotkin (1842Kropotkin ( -1921) ) [START_REF] Ferretti | Elisée Reclus, pour une géographie nouvelle[END_REF]2018;[START_REF] Kinna | Kropotkin, Reviewing the Classical Anarchist Tradition[END_REF][START_REF] Springer | The anarchist roots of geography: towards spatial emancipation[END_REF], and the widely acknowledged role of present-day rediscovery of anarchist geographies in fields such as historical geography [START_REF] Keighren | History and philosophy of geography II: The excluded, the evil, and the anarchic[END_REF]. What deserves further reflection is that, as anticipated, most of the anarchist tradition, including Reclus and Kropotkin, explicitly refused the label of utopianism, considering that anarchist activists should strive for concrete societal modifications rather than build 'Icarias' outside the real world, including utopian communities overseas [START_REF] Ferretti | They have the right to throw us out: the Élisée Reclus' Universal Geography[END_REF]. Nevertheless, according to Ruth Kinna, some tendencies in anarchism, normally considered as 'romantic' ones, adopted the idea of utopia as synonymous with the freed society that they dreamt. Kinna's argument is that, between anarchist utopianism and anti-utopianism, 'the differences have been exaggerated' (Kinna 2009, 222). This matches Maria Luisa Berneri's distinctions between 'authoritarian' and 'libertarian' utopias as I explain below.

My argument is twofold. First, as Berneri did, geographies of utopia should distinguish between, on the one hand, an imperial model which characterised the ultramarine expansion in the early Modern Epoch, reflected by the first self-declared Utopia (1516) of this kind by Thomas More [START_REF] Martínez | El impacto del Nuevo Mundo en la invención de Utopía de Tomás Moro[END_REF]2019) and, on the other, a pluralist and complex inspiration for more recent prefigurative thinking associated with the idea of social progress. Second, utopia is a pertinent object of study at the same time for anarchist geographies and cultural and humanistic (or neohumanistic) approaches to geography as a literary genre, a conceptual device for investigating cultures of empire but also as a potential tool for prefiguration and social action. Considering utopias in their places and in their historical contexts is essential: as a corollary to my first argument, I contend that there is generally (but not necessarily) a conceptual difference between utopias located in imaginary places overseas, linked to imperial imagination, and utopias which try to locate possible futures in concrete places, associated with proposals of social modifications.

In this sense, it is worth distinguishing between given models (often corresponding to non-places or imaginary places) and prefiguration, which generally implies experiences of in-becoming transformation of existing places and social spaces. This case can fit Howard Segal's general definition of utopias as ideas that 'frequently seek not to escape from the real world but to make the real world better' (Segal 2012, 7), as well as his arguments about the 'resurgence of utopia' through technology in virtual communities. Yet, these distinctions are only working tools to provide keys for reading the spaces of Utopia and should never been taken as Manichean ways to distinguish between 'good ' and 'bad' utopias. This paper also extends literature on humanistic geographies and in general on the relation between geography and the 'humanities'. Recently emerged, the field of study called 'geohumanities' is considered at the same time as a perspective potentially affecting environmental humanities (Castree, in [START_REF] Hawkins | What might GeoHumanities do? Possibilities, practices, publics, and politics[END_REF], cultural and historical geographies (Daniels, in [START_REF] Hawkins | What might GeoHumanities do? Possibilities, practices, publics, and politics[END_REF] and participatory research [START_REF] Hawkins | What might GeoHumanities do? Possibilities, practices, publics, and politics[END_REF]. It is generally considered as an interdisciplinary approach (Mitchell, in [START_REF] Hawkins | What might GeoHumanities do? Possibilities, practices, publics, and politics[END_REF] accompanying 'the spatial turn in the humanities' (Cresswell et al. 2015, 6). Several of these authors also variously refer to the 'classical' tradition of humanistic geographers [START_REF] Buttimer | The human experience of place and space[END_REF][START_REF] Buttimer | Geography and the human spirit[END_REF][START_REF] Ley | Humanistic geography, prospects and roblems[END_REF] who claimed against technocracy and recovered the idea of human freedom from early humanist thinkers like Erasmus, a philosopher who was likewise evoked by Berneri. Recent scholarship has demonstrated the political, ethical and scholarly interest of early anarchist geographers for 'humanism', 'humanitarianism' and 'humane science' against the various utilitarian and amoral approaches to the production of knowledge [START_REF] Ferretti | Anarchy and geography. Reclus and Kropotkin in the UK[END_REF]. This parallels the rediscovery of the connection between 'humanistic' approaches to geography fostered by Anne Buttimer (1938Buttimer ( -2017) ) and what was considered as 'radical geography' in the 1970s and 1980s (Ferretti 2019a). This paper extends this scholarship by addressing literary and historically approaches to utopia as an object for geographical inquiry, paying attention to the humane and humanistic sides of performative utopias.

Yet, Utopia remains an ambivalent and complex concept, associated with political radicalism but also with the imperial pasts of geography and cartography. The notion of utopia was mobilised all along the history of the socialist movements, and one of its most frequent uses is that of a label to despise options considered as unrealistic. This was typically done in classical Marxism, where all the variegated socialistic schools which did not match the so-called 'scientific' canons of Marx and Engels were equally labelled as 'utopian socialism'. In 1967, Herbert Marcuse argued for the 'end of utopia' defining it as an impossible political project, but also as a problem with which the socialist movement had already got rid at that moment, when 'the material and intellectual forces for the transformation are technically at hand' (Marcuse 1970, 47). For Marcuse, this would have allowed overcoming the objective hindrances that conservative forces put on the 'way for revolution'. A common point between this notion and the anarchist tradition might be the fact that, as anticipated, activists committed to build a different society generally don't like to be defined as 'utopians' (with some exceptions, indeed).

While an interdisciplinary field of 'utopian studies' is well established, including a dedicated journal2 and recent monographs [START_REF] Bell | Rethinking Utopia: place, power, affect[END_REF], geographers generally seem to engage rather occasionally with utopia, albeit in a rich and heterogeneous range of ways. For most geographers, the concept of Utopia was an invention of modern Europe and participated in the formation of imperial geographical imaginaries [START_REF] Gregory | Geographical imaginations[END_REF] and to the establishment of the idea of territory and territorial state [START_REF] Elden | The birth of territory[END_REF]. According to Edward Said, imagined geographies were a crucial part of the construction of an essentialising orientalist imagination underpinning Western imperial interests in 'Oriental' lands, that is a key drive for colonial expansion [START_REF] Said | Orientalism[END_REF]. According to Felix Driver, the complex field which can be defined as 'imaginative geographies' goes beyond the individual perception, being an 'inherently social' (Driver 2014, 235) notion of which imagery and mappings are fundamental parts. Crucially, for Driver, these representations 'matter', that is, they are designed to influence people's behaviour and, albeit very varied, they can still serve 'the pursuit of geopolitical power' (Driver 2014, 246). In Michel Foucault's works, discussions on 'other spaces' mark the philosopher's interest for geography and space: utopia finds a place in this elaboration as the definition of places which lack material reality but are strictly connected with existing societies. For Foucault, utopias 'are localisations without a real place. They are the localisations which have a general relation of direct or inverse analogy with the actual space of society. This can be the actual society perfectioned or its contrary, but these utopias are spaces fundamentally and essentially irreal' (Foucault 2004, 14-15). Conversely, for Foucault, heterotopias are material spaces which define an alterity and can be also considered as 'utopias actually realised' (Foucault 2004, 15), famously corresponding with spaces of segregation, temporary or permanent, for different people and 'other' social relations. If one wants to use a Foucauldian language, one might argue that Berneri considered the totalitarian regimes of the twentieth century as heterotopias associated with the 'mainstream' utopian tradition of modern Europe as I explain below.

On a slightly different note, many contemporary geographers consider utopia as a conceptual tool to reflect on urbanism and planning, in opposition to the concept of 'dystopia', i.e. a condition which 'has come to dominate Western thinking about society ... The calamities of nineteenthcentury capitalism gave rise to powerful critiques and alternatives, from communism to socialdemocracy, but the implementation of socialist and social democratic ideals in post-war decades, in turn, created its own "degenerate utopias"' (Baeten 2002a, 3). This degeneration is identified with the social and environmental disasters of neoliberalism. The editor of a special Geografiska Annaler issue on geographies of utopianism, Guy Baeten argues that, despite the proclaimed end of socialistic utopias (not in the same sense as Marcuse intended it), 'a considerable number of classical texts on modern urbanism would contain visionary elements about the virtues of urban life and would celebrate urban modernity, albeit with certain reservations' (Baeten 2002b, 148).

Drawing upon the tradition of 'utopian' urbanism by Patrick Geddes and Lewis Mumford, Baeten argues that the idea of foreseeing futures remained embodied in urban thinking also after the end of big Utopias intended as metanarratives. 'What binds utopianism, dystopianism and urbanism together in the twentieth century is an ambivalent relation: most leading commentators on the modern urban condition would foresee simultaneously a grand future for urban life and abhor it' (Baeten 2002b, 151).

Likewise, for David Pinder, the crisis of modernism did not imply the end of all forms of utopia, and it remains necessary 'to leave behind the authoritarianism and static projections typically associated with the concept of utopia, and to rethink the potential functions of utopian urbanism in an era all too ready to jettison the very idea of utopia' (Pinder 2002, 231). This implies criticising the processes of commodification responsible for the contemporary 'utopic degeneration' represented by 'non-places' like Disneylands and other hyperrealities [START_REF] Soja | Postmodern geographies: the reassertion of space in critical social theory[END_REF][START_REF] Jameson | Archaeologies of the future: the desire called utopia and other science fictions[END_REF], but also 'a rejection of authoritarian utopian schemes' (Pinder 2002, 237) to rethink utopia.

Openings in this direction could be offered by claims for a new utopian urbanism associated with the concept of hope and with geographies of affect ad affection [START_REF] Anderson | A principle of hope: recorded music, listening practices and the immanence of Utopia[END_REF].

More recently, geographers have addressed counterfactualism as a definition which can include the utopian tradition as 'worlds that might have been'. Drawing upon concepts such as Doreen Massey's 'possibility', David Gilbert and David Lambert focus on the possible 'exploration of alternative worlds' (Gilbert and Lambert 2010, 248) by geographers, providing further links between geography and the humanities, already interested in counterfactuality. According to James Kneale, the link between historical fiction, geography and utopia can be used to reflect critically on the utopian tradition and its possible outcomes. This takes inspiration from Fredric Jameson's idea that utopia, considered as synonymous with 'socialism or totalitarianism', is regaining vitality among the new social movements through a multiplicity of perspectives, including a 'multiple and open … spatialisation of the utopian impulse' (Kneale 2010, 300).

Therefore, geographies of utopia can contribute to both epistemological and activist reflections on these topics. Kneale concludes that: 'If counterfactuals are an essential part of the practice of history, and if counterfactual histories can be radical and utopian, then it seems ridiculous to try to avoid them just because a few of these histories seem frivolous or reactionary. And … rejecting them suggests an unwillingness to take historical geography seriously' (Kneale 2010, 304). Thus, Utopia can be a way for linking geography and humanities, including through fiction and performances which especially interest scholarship on both geo-humanities and nonrepresentational approaches [START_REF] Thrift | Non-representational theory: space, politics, affect[END_REF]. Moreover, as observed by Pinder, several critical and feminist authors have started to reconsider utopia as an option for thinking a different world, one where these authors confer 'a central role to desire and to moving beyond present limits into spaces and futures that are necessarily as yet unknown' (Pinder 2009, 796).

In the first part of my paper, I address Marie-Louise Berneri's transnational trajectory drawing upon her archives and relevant literature, especially focussing on her writings on World War Second and the relations between 'democracies' and 'totalitarianisms', a discussion which is indispensable to understand Journey Through Utopia. In the second part, I discuss Berneri's work on Utopia in the context of anarchist readings of prefigurative utopian spaces and their possible openings for geography, comparing her ideas with the work of one of her main inspirations, Lewis Mumford. I conclude arguing for the relevance of utopia for geography, and of geography for building new, and decolonized, ideas on utopias.

Maria Luisa Berneri: a transnational trajectory

Maria Luisa Berneri was born in 1918 to a couple of Italian anarchist intellectuals, Giovanna Caleffi (1897Caleffi ( -1962) ) and Camillo Berneri. At the age of eight, she had to follow her parents in their exile in France due to fascist persecutions in Italy [START_REF] Madrid Santos | [END_REF]. According to recent scholarship, anarchism cannot be understood within national frames given the traditional mobility of its activists for reasons of political persecution, economic migration or political propaganda, nor could it be understood overlooking its places, concrete practices and contexts. As anarchism is irreducible to theory alone, relational, contextual and spatial-sensitive readings are necessary to make sense of its complex tradition [START_REF] Bantman | Reassessing the transnational turn: scales of analysis in anarchist and syndicalist studies[END_REF][START_REF] Ferretti | Anarchy and geography. Reclus and Kropotkin in the UK[END_REF][START_REF] Turcato | Making Sense of Anarchism: Errico Malatesta's Experiments with Revolution, 1889-1900[END_REF].

While Italian-speaking anarchism is considered as one of the first and exemplar cases in this line of research [START_REF] Turcato | Italian Anarchism as a Transnational Movement, 1885-1915[END_REF], the Berneri family can be considered as an outstanding case in these transnational, cosmopolite and multilingual trajectories, both biographically and intellectually.

Exiled in and often expelled from Switzerland, France, Belgium, Netherlands, Luxembourg, Germany and finally Spain from 1926 to 1936, Camillo Berneri was defined as 'the most expelled anarchist in Europe'. Committed to fight Nazi-fascism but to oppose likewise capitalism and all forms of authoritarianism, he was murdered in Barcelona in May 1937 by Stalinist agents, while he was fighting for the defence of the social revolution accomplished by the Spanish proletariat during the Civil War from 1936 to 1939 [START_REF] Breitbart | Impressions of an anarchist landscape[END_REF][START_REF] Carrozza | Dizionario biografico degli anarchici italiani[END_REF][START_REF] Madrid Santos | [END_REF].

The Berneris' rich archives survive in the Berneri-Chessa Archive in Reggio Emilia, containing the materials which were saved by Giovanna Caleffi during the storm of World War Second and were later the object of many studies and publications in Italy (Berneri 1984 and2001;[START_REF] Carrozza | Berneri e il fascismo. Problemi e chiavi di lettura[END_REF][START_REF] De Maria | Camillo Berneri tra anarchismo e liberalismo[END_REF]. Marie-Louise's original papers survive in the Vernon Richards collection at the Amsterdam International Institute of Social History, while copies are held in Reggio. The Amsterdam folders containing her correspondences and work materials have been especially consulted for writing this paper. Adolescents in France, both Maria Luisa and her sister Giliana Berneri (1919Berneri ( -1998) ) became anarchist activists [START_REF] Chessa | Giliana Berneri[END_REF], and the violent death of their father only strengthened their anti-authoritarian ideas. By the date of this tragic event, Maria Luisa was already in London, where she had known Vero Recchioni (1915Recchioni ( -2001)), the son of Emidio Recchioni (1864Recchioni ( -1934)), who was raised like her in a family of Italian transnational anarchists emigrated to England. Relatedly, she resided at that moment in Soho, 'in the same building where Emma Goldman lived' (Chessa and Sacchetti 2003, 151). Vero, known under his pen name of Vernon Richards, became Maria Luisa's husband, and together with him she became one of the leading figures of British anarchism, among the continuators of Freedom Press and the editors of Spain and the World, War Commentary and Freedom, and one of the founders of the London group of an early 'Anarchist Federation' in 1942, as she related in a letter to an unnamed correspondent. 3 Meanwhile, she got acquainted with Colin Ward , George Woodcock (1912-1995), Herbert Read (1893-1968) and some elderlies of the Kropotkin's Freedom group, like Lilian Wolfe .

In 1945, the group of War Commentary was charged with conspiracy for their antimilitarist propaganda (Di Paola 2013) and its members remained in prison for several months. Ironically, Maria Luisa was not imprisoned because, according to a sexist law, a married woman could not be charged of conspiracy if she had done this with her husband (eventually Richards): immediately freed, she led a campaign of solidarity which saw the mobilisation of intellectuals such as George Orwell and Bertrand Russell supporting the imprisoned anarchists [START_REF] Sacchetti | Eretiche: il Novecento di Maria Luisa Berneri e Giovanna Caleffi[END_REF]. Meanwhile, she continued War Commentary and the Freedom Press activities, closely corresponding with her husband in jail. 4 As shown by her archives, Marie-Louise did not only exchange intensely with British anarchists: she also networked with international activists including Italian, Spanish, French and North-American anarchists. From 1939, she served as a liaison for Spanish anarchists in exile after their defeat in the Civil War, as suggested by a correspondence from CNT leader Diego Abad de Santillan, then exiled in Argentina. 5 Later, Berneri actively militated in favour of the Committees for Spanish Republicans and especially for the prisoners in the Chorley Camp in Lancashire, who were brought to Britain as war prisoners in 1944 and kept in despicable conditions by the 'Allies' despite being experimented anti-fascists [START_REF] Cleminson | Spanish anti-fascist 'prisoners of war' in Lancashire, 1944-46[END_REF]). To give an example of what international solidarity meant among these migrants and exiles, in 1945 a Spanish activist confined in Chorley wrote to Maria Luisa remembering the figure of her father and his struggle in Spain, and calling the addressee an: 'Orphan of our war'. 6 In the early 1940s, letters from Herbert Read alluded to a Maria Luisa's proposal for publishing in English extracts from Errico Malatesta's (1853Malatesta's ( -1932) ) writings, confirming the clear insertion of her work in the tradition of Malatestian anarchism. 7 In addition, among her papers one can find the sketch of a project for publishing the 'Complete Works of Peter Kropotkin'. 8 If bringing the international anarchist tradition into the Anglophone world was one of her priorities, Maria Luisa conceived her role of cultural transferor also in the other direction, as shown by her Marie-Louise's prolific output of the 1940s shows the depth of her intellectual interests and the effectiveness of a publishing philosophy that she explicitly defined as the need to 'go to the people with simple, clear ideas' (Various Authors 1949, 20), a programme performed through works like her pamphlet denouncing the condition of workers under Stalin's dictatorship in Russia [START_REF] Berneri | Workers in Stalin's Russia[END_REF], which was distributed in 'two printings, totalling ten thousand copies' (Various Authors, 1949, 22). Many of the articles that Maria Luisa wrote for the anarchist press in the 1940s were collected in the anthological book Neither East nor West, reproducing writings appeared in War Commentary and Freedom from 1939 to 1948. These papers account for the difficult task that Maria Luisa's generation of transnational anarchists faced in the 1940s, that is resisting the triple grip of the overwhelming forces of Fascism, Stalinism and Western 'democracies', trying to put forward a suitable alternative and updating the anarchist tradition in order to deal with the new political and social realities. Berneri did not live long enough to see the continuation, but the history of the last 70 years showed that the anarchist movement came out severely diminished and wounded from the traumatisms of the two world wars and of the totalitarianisms, but it never disappeared. Instead, it even experienced several moments of revival, around 1968, in the 1990s and also in the last few years, at least at the level of culture and scholarship. Therefore, it is worth rediscovering these Berneri's writings because they can still provide insights on the political and ethical grounds which should inspire a libertarian alternative for a society increasingly shaped by conflicts, divisions and barriers among different peoples.

Following her denunciations of the situation in Russia, Berneri did not indulge in concessions towards those sectors of the political Left which still hoped in the virtues of state communism.

However, her criticisms towards democracy were not less harsh. In the War's years, the antimilitarist option of the anarchists, in favour of objection and desertion, led to the judiciary pursuits mentioned above. Berneri supported these options by considering this conflict not as the 'good' war against the evil fascism, but as a clash between different forms of imperialism, equally violent, eventually 'between the old form of imperialism represented by the United States and the new represented by Germany and Japan' (Berneri 1952, 23). Berneri's writings systematically denounced the contradictions of Western democracies like the United States, exposing their despicable treatment of workers, of minorities (e.g. Afro-American communities, and even the Black troops fighting over all fronts) and of civil rights (e.g. the Sacco and Vanzetti affair). Special moral indignation was expressed by Berneri in relation to the massacres of civilians performed by the Allies' aerial 'terrorist raids' (Berneri 1952, 83) on Germany, Italy and occupied countries like France. Already in 1943, Berneri argued that these massacres were not less criminal than the Nazi ones, and were mainly designed to hit poor people, demoralising not only the 'enemy' but also potentially progressive forces which were awakening in the industrial centres of continental Europe. This was the case with Northern Italy, whose 'liberation' was relativized by Berneri considering that the fall of Fascism and the Allies' occupation did not actually entail a liberation, but a new war, this time between Stalinists and Anglo-Americans, for future influence in the area.

Horrified by the '[incomparable] horror' (Berneri 1952, 88) of the 1944 bombing of Leipzig, Berneri equally criticised the 'horrors' (Berneri 1952, 119) of the Peace Conferences in the months following the end of the war. She eventually argued that the UNO was not an instrument for peace, but an association defending imperialist interests, one where 'small nations must be kept in their place' (Berneri 1952, 140). Anticipating postcolonial scholarship and even classical anti-colonial arguments from Aimé Césaire's Discours sur le colonialisme [START_REF] Césaire | Discours sur le colonialisme[END_REF]), Berneri noticed how British (and in general European) brutality in the colonies contradicted the public representation of the 'good' democracies fighting against 'evil' totalitarianisms. For Berneri, treaties designed to get rid of these totalitarianisms were ridiculous while 'American, British, French and Dutch imperialisms hurried to take over the whip with which the Japanese Government held the Indonesian and the Indo-Chinese under subjection' (Berneri 1952, 125) to crush anti-colonial revolts ongoing in the region. The main Berneri's argument anticipating post-colonial scholarship was her explicit comparison between the Nazi crimes and the colonial ones, arguing for instance that: 'The campaign of lies and defamation which has accompanied the use of naked force in the Far East equals anything Goebbels might have engineered' (Berneri 1952, 126). This was done by mobilising 'Seaforth highlanders shooting down Indonesians [with] American bullets' (Berneri 1952, 128), with the complicity of progressive political forces like the Labour Party which, for Berneri, after coming to power performed 'an imperialist policy worthy of any Tory government' (Berneri 1952, 128). Quoting Pandit Nehru, impatient to 'throw away the yoke of slavery' (Berneri 1952, 123), Berneri exhorted colonised peoples to count only on their force to recover their freedom, and possibly on the support of international workers, launching the slogan: 'Not a soldier, not a round of ammunition not a machine gun not a plane for British intervention in Asia' (Berneri 1952, 125). were she denounced a 'totalitarian attitude adopted by the Allies' (Berneri 1952, 113) including denying freedom of expression under pretext of de-Nazification. Journey Through Utopia needs to be understood in these political and intellectual contexts, of which it is a clear extension in the field of cultural production.

Spaces of utopia: an anarchist critique

Lewis Mumford and his 'utopian stories'

In 1922, North-American urbanist and polymath Lewis Mumford (1895Mumford ( -1990) ) published his famous Story of Utopias, with an illustrated preface by historian and artist Hendrik Van Loon (1882[START_REF] Berneri | Workers in Stalin's Russia[END_REF] who likewise collaborated to the elaboration of Patrick Geddes's Valley Section (Ferretti 2017a). This book had a great importance in 'canonising' historical views of Utopias which lasted until recent 'Histories of Utopia', considering it as a concept extensible to the Antiquity (despite the definition clearly characterises modern Europe), by addressing Plato's Republic as the first classical masterpiece of the genre [START_REF] Segal | Utopias, a brief history from ancient writings to virtual communities[END_REF]). Mumford designed an optimistic concept of Utopia, drawing upon the 'positive' urban utopias which grew in late Victorian Britain under the influence of Geddes and Kropotkin, namely the 'numerous streams of utopianism that influenced urban planning, including notably those promoted by Ebenezer Howard and the Garden City movement' (Pinder 2002, 233). Notably, Mumford did not locate his possible utopia in the future as many of classical utopian authors, but in a rather timeless dimension, as witnessed by his models taken from a concrete historical past (including Reclus's and Kropotkin's idea of the medieval city as the cradle of mutual aid practices). As Baeten observed: 'Mumford was simultaneously horrified by modern urbanism and had little sympathy for the direction of technological progress. His urban ideal was, in the end, the medieval city, which had the right balance of dense neighbourhoods, identities and a broader civic unity' (Baeten 2002b, 149).

Nevertheless, the location of Mumford's utopia was geographically well defined: it was the real city understood as a place of human socialisation. Mumford's preferences went towards prefiguration of concrete spaces of utopia rather than in placing models overseas, in islands or even in other planets as the rising genre of science fiction was suggesting at that time. Written after the traumatism of the First World War, in a period characterised by social degradation and class struggle in European and North-American industrial cities, Mumford's book called for a reconsideration of Utopia as a 'human need' (Mumford 2015, 15) defined as 'reconstructed environment'. Yet, it was 'not merely a physical thing' (Mumford 2015, 18) because Utopia needed an ideal, in opposition to an epoch dominated by 'practical men'. Therefore, it is possible to consider Mumford's aim as a strongly humanistic one, in opposition to the domination of scientism which will be called later technocracy by authors akin to left/libertarian thinking such as Georges [START_REF] Gurvitch | Industrialisation et technocratie[END_REF].

According to Mumford, utopians 'should not abandon the real world' (Mumford 2015, 22). For doing that, the American urbanist discussed utopias in geographical terms, by quoting geographers or geographically-minded authors such as Kropotkin, Geddes and Herbert J. Fleure (Mumford 2015, 147). The first example of this was provided by Mumford's reading of Plato's Republic, manifestly one of the utopias that he cherished. In fact, Mumford noticed that Plato was not locating his model of society in some remote island, but in the concrete soil of Greece. This implies the possibility of a non-colonial utopian thinking, because Mumford's idea of territorial cohesion was opposed to the spatiality characterising colonial conditions, for which he provided the example of the Dutch 'possession' of Java. 'There is scarcely any conceivable way in which a Dutchman in Rotterdam, let us say, possesses the Island of Java: he does not live on the island, he is not acquainted with the inhabitants, he does not share their ideas or customs. His interest in Java, if he has an interest at all, is an interest in sugar, coffee, taxes, or missions. His state is not a commonwealth in the sense that it is a common possession' (Mumford 2015, 25). For Mumford, the alternative model, similar to Plato's Utopia, was 'what the geographer calls the Valley Section' (Mumford 2015, 26). This model, that Patrick Geddes launched taking inspiration from Elisée Reclus's geography (Ferretti 2017a), was considered as the ideal dimension for studying the geohistorical interactions between the city and its environments, performing the regional survey to realise a general social planning [START_REF] Geddes | The valley plan of civilization[END_REF]. Though his optimism clashes with the historical reality of slavery, patriarchy and imperialism existing in the Athenian society described by Plato, Mumford used geography to ground utopia in materiality, to address the concrete needs of the present, a task for which he greatly valued Geddes's notion of regional planning and Kropotkin's idea of 'mixture of town and country' (Mumford 2015, 50).

Mumford argued that the celebrated models of More's Utopia and Andreä's Christianopolis offered 'little progress' in comparison to Plato's original idea, and Bacon's New Atlantis or Campanella's City of the Sun had little insights to provide for a contemporary utopian thinking.

For Mumford, some 'fresh regions' (Mumford 2015, 77) were added to the realm of Utopia by eighteenth-century philosophers through the myth of the 'noble savage'. Yet, what mostly interested Mumford were the socialist utopias of the nineteenth century, much closer to his own concerns about contemporary industrial cities. There, Mumford anticipated some themes developed later by Berneri, namely a distinction between more and less authoritarian (and normative) utopias. For instance, he criticised Icaria, imagined by French early communist Etienne Cabet, for the 'dictatorial power' which ruled this community that he deemed 'so little like Utopia' (Mumford 2015, 105) and so much like the real world. Likewise, Looking Backward by Edward Bellamy was considered by Mumford as an authoritarian and technocratic dream, where every worker is 'a cog in the machine' (Mumford 2015, 111). On the contrary, William Morris's News from Nowhere was deemed a more interesting model, being established in a concrete geographic place, the 'Thames Valley' (Mumford 2015, 117), and inserted in a future socialistic and post-industrial Britain, where the individuals' life was not strictly regulated as in the other utopias.

According to Mumford, A Modern Utopia by H.G. Wells represented the coming back to imaginary locations, because it was placed in a new planet 'beyond Sirius' (Mumford 2015, 122) and ruled by the elite class of the 'Samurais'. For Mumford, what was definitively needed was a realistic appraisal of modern utopias in the existing society: these were the Country House, inspired by pleasant dreams of freedom but in effect a privilege for a social elite exerting a 'rapacious' (Mumford 2015, 124) exploitation of the wider community, and Coketown. This latter was based on technology, production and division of labour, but also on 'the horrible mistake of believing that all these things were good in themselves' (Mumford 2015, 145) and of neglecting 'civic life'. For Mumford, the ideal of civics was the medieval commune characterised by participation and mutual aid, a stance widely inspired by the ideas of Reclus and Kropotkin.

Conversely, for Mumford, the final outcome of Coketown and Country House was 'the collective utopia of the nation state' (Mumford 2015, 146), that Mumford criticised starting by its geographical bases and the very idea of territory. 'In the utopia of the national state there are no natural regions' (Mumford 2015, 148) nor respect for historical and cultural affinities of people, because the shape of the state is artificially produced by 'the surveyor's line' (Mumford 2015, 148). Again, Mumford matched the tradition of anarchist geographers in criticising the arbitrary nature of state and administrative regionalisation.

Finally, the main 'instrument for the nation state' (Mumford 2015, 149) was Megalopolis, the big industrial concentration where direct action and public participation to decision-making were impossible. Thus, for Mumford, it was time to overtake the 'externalism' (Mumford 2015, 163) of nineteenth century utopias, generally located in the past, in the future or in remote and imaginary countries, and to get rid of utopian 'idola' to make ideas work in the 'everyday world' (Mumford 2015, 176). For that, it was necessary to relativize science, given that 'the independence of science from human values is gross superstition' (Mumford 2015, 182). For Mumford, an important tool for linking social and scholarly action was the Outlook Tower by Patrick Geddes, likewise inspired by Reclus (Ferretti 2017a), assuming regional survey as 'a way for coming back to the real world' (Mumford 2015, 186) and for challenging the 'divorce between humanities and science' (Mumford 2015, 188) as well as the commodification of arts and knowledge. Rather than claiming for utopia, considered as an ambiguous term, Mumford preferred to introduce the notion of 'eutopia' for 'converting the idolum of eutopia into plans' (Mumford 2015, 201), among which he included the Garden City movement. Mumford finally argued for the plurality of the very concept of utopia, as a single utopia would never fit the entire humankind, and for the principle of social justice. Mumford was a critic of the utopian tradition, but he can also be considered as an enthusiastic of some of its aspects, as shown by the conclusions of his book arguing that 'our most important task now [is] to build castles in the air' (Mumford 2015, 204).

Berneri: what utopia ought to be An anarchist critique of Mumford came from Maria Luisa Berneri, whose Journey Through Utopia deserves rediscovery and recognition. This book is never quoted in Segal's work mentioned above, nor in Ruth Levitas's Utopia as a method (2013), albeit Levitas rapidly analyses Berneri's work in the Concept of Utopia considering it as 'ambivalent about the function of utopia' (Levitas 2010, 29). I would contend that the writings collected in Neither East nor West [START_REF] Berneri | Neither east nor west: selected writings[END_REF]) constitute the keys for reading Berneri's work on Utopia: albeit strongly inspired by Mumford, Berneri took some distance from the optimism of this latter. Writing in years when anarchist and progressive movements had been crushed by the Nazi-Fascist and Stalinist totalitarianisms and when the atomic bomb had already exploded, she compared many of the historical utopias with these authoritarian experiences, expressing a much more sceptical stance on the 'positive' role of science and knowledge.

Journey Through Utopia was published as a posthumous work and was prefaced by Woodcock.

Among the first differences with Mumford, one notices that Berneri presented Utopia as a problematic concept instead of pleading for it, and that she gave up the idea of historical progress which Mumford evoked. In that, Berneri somehow anticipated historical and philosophical analyses considering Utopia as the idea of imperial European modernity [START_REF] Bauman | Dialectic of modernity[END_REF][START_REF] Koselleck | Futures past: on the semantics of historical time[END_REF]). If Reinhard Koselleck has discussed the different temporalities of the 'presents' and 'futures' of the past, it is possible to consider that Berneri made a critique of early 'views on the future', one which was strongly inserted in her own present. Her book was written in the same years as Orwell 's 1984 (1948-49) and participated in the anarchist critique of 20 th century totalitarianisms and technocracy [START_REF] Gurvitch | Industrialisation et technocratie[END_REF]). An inspiration for Berneri was her friendship with Vsevolod Mikhailovich Eichenbaum (1882-1945) known as Volin, a Russian anarchist who was finishing his famous book The Unknown Revolution while he corresponded with Berneri, some months before his death in 1945. 11 A classic of anarchist literature, Volin's book told the history of the deception of Russian and Ukrainian anarchists and other revolutionaries who had been protagonists of the 1917 Revolution and were then ferociously repressed when the Bolsheviks established a dictatorial regime in the immediately following years. As Volin explained in his letters to Berneri, this editorial endeavour was first solicited by French anarchist intellectual Sébastien Faure , the editor of the Encyclopédie anarchiste (1934), who stressed the urgency of doing an alternative 'History of the Russian Revolution', 12 one which could expose its making 'not at the top but at the bottom', 13 popularising the conceptual distinction between social revolution and totalitarianism. After Volin's death, Richards and Berneri were among those who looked after the publication of the first edition of his book, printed in Paris as La Révolution inconnue. 14 By adopting Mumford's definition of 'the utopia of the nation state', Berneri first matched his ideas about the artificial character of the territories described by those classical utopias which naturalised the state as something taken for granted rather than a social and historical construction.

For Berneri, Enlightenment authors like Etienne Morelly, 'instead of trying to discover the laws of nature … preferred to invent them' (Berneri 1950, 4). Likewise, utopians tend to invent 'rational' geographies for their purposes: 'To the uniformed nation must correspond a uniform country or city. The authoritarian love of symmetry causes utopias to suppress mountains or rivers, and even to imagine perfectly round islands and perfectly straight rivers' (Berneri 1950, 5). On the one hand, this anticipated post-structuralist critiques of representation and naturalization of geographical entities. On the other, this idea matched similar critiques of mapping expressed by early anarchist geographers Reclus and Kropotkin [START_REF] Ferretti | Evolution and revolution: anarchist geographies, modernity and poststructuralism[END_REF]).

11 IISH, Vernon Richard Papers, Folder 278, letters from Volin to Marie- Louise Berneri, 8 May 1945-19 July 1945. 12 IISH, Vernon Richards Papers, 278, Volin to Marie- Louise Berneri, 11 May 1945. 13 IISH, Vernon Richards Papers, 278, Volin to Marie- Louise Berneri, 6 June 1945. 14 IISH, Vernon Richards Papers, 270, Gustave Franssen to Vernon Richards, 25 February 1948. Berneri's main argument is that a distinction was first needed between authoritarian utopias (the majority) and libertarian exceptions. She defined the first cases as 'utopias without life', arguing, as Woodcock explained in the Preface, that 'most … of these schemes must of necessity become a gruesome prison unless it is based firmly and securely on the foundation of individual freedom' (Woodcock 1950, xi). This corresponded to a humanistic critique of the idea of 'perfect society': here, utopian societies are deemed very 'boring', in opposition to real life and everyday struggles to change society. Great importance was given to daily life: like Reclus and Kropotkin, Berneri expressed some sympathy for François Rabelais's Abbaye de Thélème (1534) but considered this 'utopia', based on the motto 'do what you please', as a rather individualistic and elitist rebellion.

Albeit Berneri did not use to define herself as a feminist, she strove for revolution in personal life, also drawing upon Wilhelm Reich's work, and criticised the family as the fundamental societal basis (Berneri 1950, 32), ridiculing the rigid social and sexual moral of many utopias castigating and despising 'the joys of the table, of laughter, of poetry, of music and of love-making … the less noble pleasures of life' (Berneri 1950, 50). This matched Mumford's critique of the 'little utopia' of family.

Though Berneri presented utopia as something mainly characterising modernity, the (relatively) few pages that she dedicated to the ancient utopias clarify her explicit political use of the past. It was the case with her comparison between Plato's Republic and State communism-cumtechnocracy, while she deemed Aristotle's ideas closer to the bourgeois reformism of her days.

Conversely, for Berneri, 'the universalist ideal of Zeno, who … proclaimed the brotherhood of men of all nations, has rarely been adopted by utopian writers' (Berneri 1950, 7). Needless to say, Zeno was defined as some of a forerunner of anarchism by [START_REF] Kropotkin | Modern science and anarchism[END_REF]. Addressing modern utopian spaces such as More's Utopia, Berneri expressed her fascination for their ideas on social justice, but also her wariness of what today would be called their imaginative geographies: that is, the inventions of cities, islands and new spaces for dominating societies. For Berneri, this corresponded to the invention of the modern state, 'which became the negation of the individual' (Berneri 1950, 56) and to the pursuit of 'a bare-faced policy of expansion' (Berneri 1950, 85).

Applying a Kropotkinian historiographic framework, Berneri considered most of modern utopias as 'simulacra' of the medieval free commune. About More's Utopia, she argues that 'in a society which admits of slavery even the "free citizen" is not free; his chain is only longer than that of the slave' (Berneri 1950, 84). Berneri clearly matched a humanistic perspective, quoting Erasmus and anticipating geographical complaints on a 'human-less' science [START_REF] Ley | Humanistic geography, prospects and roblems[END_REF] by noting that, in the depictions of these monastic societies, people are 'completely inhuman in that they are incapable, or are forbidden, to have any feelings other than those dictated by certain laws' (Berneri 1950, 61).

As 'humanistic geographers' did a couple of decades later, Berneri paid close attention to people's spaces and places of life, and their concrete locations. She noticed that Tommaso Campanella identified his City of the Sun with the 1602 Republic of Calabria, trying to convince his readers that the ideal city was not located in an exotic land and to make them 'think of it as around them and themselves as its citizens' (Berneri 1950, 92). Nevertheless, Berneri was clear in condemning Campanella's eugenics as a feature that sadly anticipated twentieth-century totalitarian politics, together with all Campanella's rigid moral and sexual prescriptions. In the following pages, she made similar critiques to the Calvinist authoritarianism of Valentin Andreä and to the 'aristocratic model' provided by Francis Bacon.

According to Berneri, in the seventieth century, the only egalitarian (tough non-libertarian) model of Utopia was provided by Gerrard Winstanley's The Law of Freedom, as it 'embodies the spirit of the English revolution in its more popular ad revolutionary form' (Berneri 1950, 143). It is worth noting that Jim Mac Laughlin (2016) recently did a comparison between Winstanley's ideas and some aspects of Kropotkin's anarchism. About the eighteenth century, Berneri endorsed as a 'positive utopia' Denis Diderot's Supplément au Voyage de Bougainville, the manifesto of anticolonial Enlightenment [START_REF] Muthu | Enlightenment Against Empire[END_REF] also discussed by current scholarship on anarchism and utopianism [START_REF] Stillman | Supplément au voyage de Bougainville': steps towards an anarchist utopia[END_REF]. Despite being referred to a real place, Tahiti, Diderot's ideas were expressed under the form of fiction. Yet, according to Berneri, this was a key expression of the need for a society where the individual was granted a great autonomy and anticipated the claims of 1789 revolutionaries for the brotherhood of all peoples beyond pretentions of racial or cultural superiority. As for nineteenth-century revolutions, Berneri observed that, at that time, 'utopian became almost a term of abuse which self-styled scientific socialists were found of hurling at their opponents' (Berneri 1950, 207). As anticipated above, this was especially the case with the Marxists claiming for 'scientific socialism' and with all those who tried to dismiss early forms of socialism on the grounds of their alleged 'utopianism'. For this reason, Berneri stressed the antiutopian components of the anarchist tradition and blamed socialistic utopias such as those elaborated by Etienne Cabet and Edward Bellamy, matching Mumford's arguments and evoking the experience of totalitarianisms: 'We are not surprised to hear that books were burned in Icaria when the regime was established, and that a rigid censorship controls the production of all works of art' (Berneri 1950, 234).

The exception per excellence, on which Berneri was much more emphatic than Mumford, was News from Nowhere. 'After the stifling atmospheres of Cabet's and Bellamy's utopias, with their complicated bureaucratic machines run by an all-wise, all-pervading state, William Morris's utopian England appears like an oasis where we would like to stay, if not for ever, at least for a long time. Here we can work without having a foreman at our elbow, we can sleep without having to set the alarm clock, eat what we like and not what the experts have decided to be best suited to our constitution; we can love without having to consider tyrannical laws or a no less tyrannical public opinion; here we can dress as we like, read what we like and, above all, think what we like.

Here we can live, because we have not been catalogued and directed, but left to arrange our lives as we think fit' (Berneri 1950, 255). Giving Berneri's commitment to what is now called contextual readings, this endorsement is also due, accordingly, to the close links that William Morris shared with Kropotkin [START_REF] Ferretti | Anarchy and geography. Reclus and Kropotkin in the UK[END_REF]).

In the twentieth century, according to Berneri, utopias were declining, because totalitarianisms and technocracies had rendered some of them real: reality had overtaken fantasy, and science fiction was ridiculing the most reactionary aspects of totalitarianism. Here, Berneri anticipated somehow Segal's argument on the distinction between science fiction and utopia, considering science fiction rather as social critique than social prefiguration (Segal 2010, 8). In her book's conclusion, Berneri praised the authors who had fought for an 'anti-utopian world. In France Jean-Paul Sartre, André Breton and Camus, in America Henry Miller …; Catholics like Eric Gill and Georges Bernanos; sociologists and biologists like Lewis Mumford and Patrick Geddes; novelists like E.M. Forster, Rex Warner and Graham Greene, have all fought the battle of the individual and against the state' (Berneri 1950, 313). Nevertheless, Berneri nuanced this anti-utopianism by mentioning Mumford's Utopias and Ethel Mannin's Bread and Roses, whose authors 'continue to assert the will to Utopia, and echo Oscar Wilde's famous remark: "A map of the world that does not include Utopia is not worth even glancing at, for which it leaves out the one country at which Humanity is always landing"' (Berneri 1950, 308). Finally, for Berneri, utopias were not only plans of technocrats, as 'they have also been the living dreams of poets' (Berneri 1950, 317). These arguments match the quest for a humane and humanistic perspective by past and to-day anarchist geographies questioning utilitarianism in science and academy to perform experiences valuing voluntarism and subjectivity [START_REF] Ferretti | Anarchy and geography. Reclus and Kropotkin in the UK[END_REF][START_REF] Springer | Earth Writing[END_REF].

Conclusion: decolonising utopia

On the one hand, this paper has confirmed the extent on which a geographical reflection underpins the anarchist tradition, and how much this tradition should be understood in its places, contexts and concrete practices through relational, transnational, multilingual, contextual and spacesensitive readings. On the other, geography owed to anarchism some of its earliest thinkers and can still gain in refreshing its traditional 'cross-pollination' with anarchist thought: the connections between anarchism and the spatiality of utopia can extend not only current scholarship in anarchist geographies. It can also provide insights to the geo-humanities and to the rediscovery of humanistic geographies and their radical charge of contestation to technocracies and to the dogmatisms of 'quantitative' science and 'modelling' approaches (Ferretti 2019a). Marie-Louise Berneri's remarks on the de-humanisation of spaces in utopian thinking all along what is called the 'modern era', in this case from Thomas More to the twentieth century, chime with successive Anne Buttimer's critiques of 'the dance macabre of materialistically motivated robots' (Buttimer 1993, 47) in the academy and beyond. This has for sure the potential of expanding the field of geographical enquiry and to strengthen geography's relation with the humanities. This paper has also shown that the anarchist tradition originally reflects the ambivalence which characterised the idea of utopia in all Western thinking. On the one hand, utopia was functional to ultramarine exploration, mapping and imperial geographic imaginaries. On the other, utopia is a reservoir of ideas in performative practices for what Levitas calls 'a definition of utopia in terms of desire ... analytic rather than descriptive' (Levitas, 2013, xiii), presenting the advantage of being 'a holistic method' (Levitas 2013, xviii). After the end of the big utopias, 'a more open, dynamic and transformative utopianism' (Beaten, 2002) seems to be possible, in a performative and even non-representational way, assuming that this kind of utopia is a self-productive one and not the abstract drawing of a model to be applied. The importance of the distinction between libertarian and authoritarian approaches is key to the political relevance of ideas on utopia: humane elaborations such as that of the anarchist critiques of utopias by Mumford and Berneri still can be considered as important insights to inform anarchist action.

It is clearly necessary to contextualise Berneri's work, inspired by the need for what Lucien Febvre defined 'organising the past for the present' (Febvre, 1953, 438) but also matching the tradition of early anarchist geographers in a reflection on spaces of social prefiguration and challenges to authoritarian imaginations. Mumford can be considered as a follower of the same tradition for his references to Kropotkin and his close relationship with Geddes, while Berneri's work was also an outcome of the tradition of Italian anarchism, from the Risorgimento's federalists such as Carlo Cattaneo (1801-1869), Carlo Pisacane (1818-1857) and Giuseppe Ferrari (1812-1876) to the communist and organisational anarchism represented by figures such as Malatesta, Luigi Fabbri (1877-1935) and her own father Camillo [START_REF] Ferretti | Transnational anarchism against fascisms: subaltern geopolitics and spaces of exile in Camillo Berneri's work[END_REF].

Further research should be done on possible non-western notions of utopia, a point sketched by some of the authors mentioned above, one that still needs further clarification.

  correspondences with Italian anarchist intellectual CarloDoglio (1914Doglio ( -1995) ) about possible Italian editions of works by Read and Woodcock. 9 A long 1943 letter from Armando Borghi (1882-1968) the leader of the Unione Sindacale Italiana (the Italian anarcho-syndicalist organisation corresponding to the Spanish CNT) during the 'Red Biennial', then exiled in North America, demonstrates Marie-Louise's efforts to reconstruct the history of the Italian anarchist movement before Fascism and to save its patrimony of ideas and practices to the public memory. 10 This programme was interrupted by Marie-Louise's sudden death in April 1949 for an illness, at the age of only 31 (Various Authors 1949). The centenary of her birth was recently celebrated in a cover of Anarchist Studies (Croswell 2018). As discussed by Matthew Adams, several themes from her work are attracting interest in current scholarship, including the strong personal and intellectual influence that she exerted in the 'conversion' to anarchism of George Woodcock, distinguished historian of the anarchist movement and Kropotkin's biographer (Adams 2015), as well as her commentaries on sexuality and psychology in Wilhelm Reich's work (Adams 2018).

Fig. 1

 1 Fig. 1 Marie-Louise Berneri. Source: https://commons.wikimedia.org/wiki/File:Berneri_ML.jpeg

  IISH, Vernon Richards Papers, 269 and 274, letters from Marie-Louise Berneri to Vernon Richards. 5 IISH, Vernon Richards Papers, 268, Diego Abad de Santillan to Marie-Louise Berneri, 23 November 1939. 6 IISH, Vernon Richards Papers, 269, Gregorio Segura to Marie-Louise Berneri, 4 October 1945. 7 IISH, Vernon Richards Papers, 268, Herbert Read to Marie-Louise Berneri, 23 March 1941.

4 8 IISH, Vernon Richards Papers, 270, 'Complete Works of Peter Kropotkin'.

An early version of this paper was presented at the session 'Where Next? Historical Geographies of the Future' at the RGS-IBG International Conference in

Thanks to all those who asked questions and gave suggestions, especially the panel's convenors Mike Heffernan and Jake Hodder. Many thanks also to Carolina Martínez (CONICET) for our conversations on utopia and modernity.

Utopian Studies, https://www.jstor.org/journal/utopianstudies

Amsterdam, International Institute of Social History (hereafter IISH), Vernon Richards Papers, 268, Marie-Louise Berneri to anonymous, 5 May 1945.

IISH, Vernon Richards Papers, 269, Marie-Louise Berneri to Carlo Doglio, 22 November 1947.

IISH, Vernon Richards Papers, 268, Armando Borghi to Marie-Louise Berneri [1943[START_REF] Berneri | Workers in Stalin's Russia[END_REF].


Phuc Ngo

Yukiko Kenmochi

Isabelle Debled-Rennesson

Nicolas Passat

Convexity-preserving rigid motions of 2D digital objects

Keywords: digital rigid motion, digital convexity, half-plane representation, concavity tree, quasi-regularity

published or not. The documents may come

Introduction

Rigid motions (i.e., transformations based on translations and rotations) in R n are well-known topology-and geometry-preserving operations. They are frequently used in image processing and image analysis. Due to digitization effects, these important properties are generally lost when considering digital images defined on Z n , as illustrated in Fig. 1.

A part of these problems, topological preservation, was studied for 2D images under rigid transformations [START_REF] Ngo | Topology-preserving rigid transformation of 2D digital images[END_REF]. In a similar context, we investigate the geometrical issue, in particular convexity, in this article.

Convexity plays an important role in geometry. This notion is well studied and understood in the continuous space. A convex object in R n is defined as a region such that the segment joining any two points within the region is also within the region. Nevertheless, this notion is not adapted to Z n , due to the digitization process required to convert continuous objects to digital objects, as illustrated in Fig. 2.

In the literature, several definitions have been proposed for convexity defined on Z n , namely digital convexity, such as MP-convexity [START_REF] Minsky | Perceptrons: An Introduction to Computational Geometry[END_REF], S-convexity [START_REF] Sklansky | Recognition of convex blobs[END_REF], Dconvexity [START_REF] Kim | On the cellular convexity of complexes[END_REF], H-convexity [START_REF] Kim | On the cellular convexity of complexes[END_REF]. The latter will be used in this article due to its simplicity and usefulness. Roughly speaking, a digital object X ⊂ Z 2 is H-convex if its convex hull does contain exactly the points of X (see Fig. 2(c-d)).

In this article, we propose a method allowing us to preserve the H-convexity of digital objects under any rigid motions. Such a method relies on a representation of an H-convex digital object by half-planes. The rigid motion is then applied on such half-plane representation and followed by a digitization. Furthermore, we investigate the condition under which the H-convexity of digital objects is preserved by arbitrary rigid motions. The paper is organized as follows. After recalling in Sec. 2 basic notions about rigid motions on digital objects, Sec. 3 presents a representation of digital object, well-adapted to H-convexity recognition. A convexity-preserving method for digital objects under rigid motions is then proposed in Sec. 4. Section 5 investigates the problem of rigid motions of non-convex digital objects and opens perspectives summarized in the conclusion.

2 Digital objects and rigid motions on Z 2

Objects and rigid motions on R 2

Let us consider an object X in the Euclidean plane R 2 as a closed connected finite subset of R 2 . Rigid motions on R 2 are defined by a mapping:

T : R 2 → R 2 x → Rx + t (1)
where R is a rotation matrix and t ∈ R 2 is a translation vector. Such bijective transformation T is isometric and orientation-preserving. Thus, T(X) has the same shape (i.e., the same geometry and topology) as X.

Digitization of objects and topology preservation

As the digitization process may cause topological alterations, conditions for guaranteeing the topology of shape boundaries have been studied [START_REF] Latecki | Preserving topology by a digitization process[END_REF][START_REF] Pavlidis | Algorithms for Graphics and Image Processing[END_REF].

Definition 1. An object X ⊂ R 2 is r-regular if for each boundary point of X, there exist two tangent open balls of radius r, lying entirely in X and its complement X, respectively.

This notion, based on classical concepts of differential geometry, establishes a topological link between a continuous shape and its digital counterpart as follows.

Proposition 1 ([20]). An r-regular object X ⊂ R 2 has the same topological structure as its digitized version

X ∩ Z 2 if r ≥ √ 2 2 .
Note that the digitization defined by the intersection of a continuous object X and Z 2 is called Gauss digitization [START_REF] Klette | Digital Geometry: Geometric Methods for Digital Picture Analysis[END_REF]. It was shown that the digitization process of an r-regular object yields a well-composed shape [START_REF] Latecki | Preserving topology by a digitization process[END_REF], whose definition relies on the following concepts of digital topology (see e.g., [START_REF] Klette | Digital Geometry: Geometric Methods for Digital Picture Analysis[END_REF]). Given a point p ∈ Z 2 , the k-neighborhood of p is defined by N k (p) = {q ∈ Z 2 : pq ≤ 1} for k = 4 (resp. 8) where = 1 (resp. ∞). We say that a point q is k-adjacent to p if q ∈ N k (p) \ {p}. From the reflexive-transitive closure of this k-adjacency relation on a finite subset S ⊂ Z 2 , we derive the k-connectivity relation on S. The k-connectivity relation on S is an equivalence relation. If there is exactly one equivalence class for this relation, namely S, then we say that S is k-connected.

Definition 2 ([16]

). A finite subset S ⊂ Z 2 is well-composed if each 8-connected component of S and of its complement S is also 4-connected. This definition implies that the boundary4 of S is a set of 1-manifolds whenever S is well-composed. As stated above, there exists a strong link between r-regularity and well-composedness.

Proposition 2 ([16]). If an object X ⊂ R 2 is r-regular with r ≥ √ 2 2 , then X ∩ Z 2 is well-composed.
We define a digital object X as a connected finite subset of Z 2 . In the context of topological coherence of shape boundary, we hereafter assume that X is wellcomposed, and thus 4-connected.

Problem of point-by-point rigid motions on Z 2

If we simply apply a rigid motion T of Eq. ( 1) to every point in Z 2 , we generally have T(Z 2 ) ⊂ Z 2 . In order to get the points back on Z 2 , we then need a digitization operator D : R 2 → Z 2 , commonly defined as a standard rounding function. A discrete analogue of T is then obtained by T point = D • T |Z 2 , so that the discrete analogue of the point-by-point rigid motion of a digital object X on Z 2 is given by T point (X). Figure 3 illustrates some examples of T point of digital lines with different thicknesses. These examples show that the topology and geometry of digital objects are not always preserved, even when the initial shapes are very simple. If a digital line is sufficiently thick, then it preserves topology, but not always geometry. This led us to consider a digital counterpart of regularity condition for guaranteeing topology during point-by-point rigid motions on Z 2 [START_REF] Ngo | Topology-preserving rigid transformation of 2D digital images[END_REF]. However, finding rigid motions on Z 2 preserving geometry is still an open problem. In this article, we focus on one important geometrical property, namely convexity. We present a new method for rigid motions on Z 2 that preserves convexity.

Digital convexity

In R 2 , an object X is said to be convex if and only if for any pair of points x, y ∈ X, every point on the straight line segment joining x and y, defined by

[x, y] = {λx + (1 -λ)y | 0 ≤ λ ≤ 1},
is also within X. This continuous notion, however, cannot be directly applied to digital objects X in Z 2 since for p, q ∈ X, we generally have [p, q] ⊂ Z 2 . In order to tackle this issue, several extensions of the notion of convexity from R 2 to Z 2 have been proposed.

Definitions

First of all, we introduce the definition of Minsky and Papert [START_REF] Minsky | Perceptrons: An Introduction to Computational Geometry[END_REF], called MPconvexity, which is a straightforward extension of the continuous notion.

Definition 3. A digital object X is MP-convex if for any pair of points p, q ∈ X, ∀r ∈ [p, q] ∩ Z 2 , r ∈ X.
Sklansky proposed a different definition based on digitization process [START_REF] Sklansky | Recognition of convex blobs[END_REF], called S-convexity.

Definition 4. A digital object X is S-convex if there exists a convex object X ⊂ R 2 such that X = X ∩ Z 2 .
Later, Kim gave a geometrical definition that is based on convex hull [START_REF] Kim | On the cellular convexity of complexes[END_REF], named H-convexity. The convex hull of X is defined by

Conv(X) = x ∈ R 2 x = |X| i=1 λ i p i ∧ |X| i=1 λ i = 1 ∧ λ i ≥ 0 ∧ p i ∈ X . Definition 5. A digital object X (connected, by definition) is H-convex if X = Conv(X) ∩ Z 2 .
Kim also showed the equivalence between MP-convexity and H-convexity for 4-connected digital objects [START_REF] Kim | On the cellular convexity of complexes[END_REF]Theorem 5]. Similar results under the assumption of 8-connectivity can be found in [START_REF] Eckhardt | Digital lines and digital convexity[END_REF] via the chord property, which relates the MP-and H-convexities to another digital convexity notion based on digital lines, called D-convexity [START_REF] Kim | Digital straight lines and convexity of digital regions[END_REF]. On the other hand, discussing the relation between Sconvexity and H-convexity requires some conditions. An element p of a digital object X is said to be isolated if |N (p) ∩ X| ≤ 1. Under the condition that X has no isolated element, it was proven that X is H-convex if and only if X is S-convex [START_REF] Kim | On the cellular convexity of complexes[END_REF]Theorem 4]. A detailed description of various notions of digital convexity can be found in [START_REF] Cristescu | Non-Connected Convexities and Applications[END_REF]Chapter 9]. In this article, we adopt the H-convexity notion, as it allows us to propose a method for rigid motions of digital objects which preserves H-convexity thanks to the half-plane representation (see Sec. 4).

It should be also mentioned that a similar property to the intersection property known from ordinary convex sets, is preserved; a proof based on an approach of the chord property can be found in [5, Corollary 3.5.1].

Property 1 Let X and Y be two digital objects. If X and Y are H-convex and

X ∩ Y is connected, then X ∩ Y is H-convex.
Let us remark that convexity does not imply connectivity in Z 2 by contrast to R 2 . This is a reason why an additional connectivity condition is reasonable in the discrete setting.

Digital half-plane representation

If a digital object X consists of more than two elements, then the convex hull Conv(X)5 is a convex polygon whose vertices are some points of X. As these vertices are grid points, Conv(X) is thus represented by the union of closed halfplanes with integer coefficients such that where R(P ) is the minimal set of closed half-planes that constitute a convex polygon P and each H is a closed half-plane in the following form:

Conv(X) = H∈R(Conv(X)) H,
H = {(x, y) ∈ R 2 | ax + by + c ≤ 0, a, b, c ∈ Z, gcd(a, b) = 1}. ( 2
)
Note that the integer coefficients of H are uniquely obtained by the pairs of the corresponding consecutive vertices of Conv(X), denoted by u, v ∈ Z 2 , which are in the clockwise order, such that (a, b) =

) = v -u ∈ Z 2 .
Therefore, from Def. 5, if X is H-convex, then we have

X = H∈R(Conv(X)) H ∩ Z 2 = H∈R(Conv(X)) H ∩ Z 2 (3)
where each H∩Z 2 is called a digital half-plane. It is obvious that any digital halfplane is H-convex. An example of an H-convex digital object with its half-plane representation is illustrated in Fig. 4.

Verification and recognition algorithms

To determine if a digital object is H-convex, several discrete methods were proposed with different approaches. In particular, they are based on the observation of the 8-connected curve corresponding to the inner contour of digital object.

In [START_REF] Debled-Rennesson | Detection of the discrete convexity of polyominoes[END_REF], a linear algorithm determines if a given polyomino is convex and, in this case, it returns its convex-hull. It relies on the incremental digital straight line recognition algorithm [START_REF] Debled-Rennesson | A linear algorithm for segmentation of digital curves[END_REF], and uses the geometrical properties of leaning points of maximal discrete straight line segments on the contour. The algorithm scans the contour curve and decomposes it into discrete segments whose extremities must be leaning points. The tangential cover of the curve [START_REF] Feschet | Optimal time computation of the tangent of a discrete curve: Application to the curvature[END_REF] can be used to obtain this decomposition. Then, the half-planes corresponding to Eq. ( 3) are directly deduced from the characteristics of discrete segments obtained in the curve decomposition. Similar approaches [START_REF] Dorksen-Reiter | Convex and concave parts of digital curves[END_REF][START_REF] Roussillon | Faithful polygonal representation of the convex and concave parts of a digital curve[END_REF] were used to decompose a discrete shape contour into a faithful polygonal representation respecting convex and concave parts.

On the other hand, a combinatorial approach presented in [START_REF] Brlek | Lyndon + Christoffel = digitally convex[END_REF] uses tools of combinatorics on words to study contour words: the linear Lyndon factorization algorithm [START_REF] Duval | Factorizing words over an ordered alphabet[END_REF] and the Christoffel words. A linear time algorithm verifies convexity of polyominoes and can also compute the convex hull of a digital object. It is presented as a discrete version of the classical Melkman algorithm [START_REF] Melkman | On-line construction of the convex hull of a simple polyline[END_REF]. This latter can also be used to compute the convex hull of a digital object, so that the half-planes of R(Conv(X)) of a digital object X are then deduced from the consecutive vertices (see Eq. ( 2)). Note that R(P ) is the minimal set of halfplanes whose support lines are the edges of a convex polygon P . Then, the H-convexity is tested with elementary operations (see Eq. ( 3)).

Convexity-preserving rigid motions of digital objects

In order to preserve the H-convexity of a given H-convex digital object X, we will not apply rigid motions to each grid point of X, as discussed in Sec. 2.3, but to each half-plane H of R(Conv(X)) according to Eq. ( 3). We first explain how to perform a rigid motion of a closed integer half-plane. Then, we propose a method for rigid motions of the whole H-convex digital object.

Rational rigid motions of a digital half-plane

A notion of rigid motion of a digital half-plane H ∩ Z 2 is given by:

T Conv (H ∩ Z 2 ) := T(H) ∩ Z 2 (4)
where T(H) is defined analytically as follows.

Digital objects and digital half-planes involve exact computations with integers. Thus, we assume hereafter that all the parameters R and t of rigid motions T are rational. More precisely, we consider R = 1 r p -q q p where p, q, r ∈ Z constitute a Pythagorean triple, i.e., p 2 + q 2 = r 2 , r = 0, and t

= (t x , t y ) ∈ Q 2 .
This assumption is reasonable, as we can always find rational parameter values sufficiently close to any real values (see [START_REF] Anglin | Using Pythagorean triangles to approximate angles[END_REF] for finding such a Pythagorean triple). An integer half-plane H defined by Eq. ( 2) is transformed by such T to the rational half-plane:

T(H) = {(x, y) ∈ R 2 | αx + βy + γ ≤ 0, α, β, γ ∈ Q}, (5)
whose coefficients α, β, γ are given by

α β = R a b and γ = c + αt x + βt y .
Note that any rational half-plane can be easily rewritten as an integer halfplane in the form of Eq. (2).

Rigid motions of H-convex digital objects

In the previous subsection, we showed how to transform a digital half-plane with Eq. ( 4) via Eq. ( 5). Since an H-convex digital object X is represented by a finite set of digital half-planes H, as shown in Eq. ( 3), we can define a rigid motion of X on Z 2 using its associated digital half-planes such that

T Conv (X) := T H∈R(Conv(X)) H ∩ Z 2 = H∈R(Conv(X)) T(H) ∩ Z 2 . ( 6
)
For each H ∈ R(Conv(X)), we obtain T(H) by Eq. ( 5), and then re-digitize the transformed convex polygon P = H∈R(Conv(X))

T(H).

We now show that T Conv (X) is H-convex under a condition on the transformed convex polygon P , called quasi-r-regularity. Thanks to Property 1, what we need here is to characterize convex polygons P whose re-discritization P ∩Z 2 preserves the 4-connectivity since any digital half-plane is H-convex.

Property 2 Let P be a (closed) convex polygon in R 2 . If P includes a (closed) ball of diameter √ 2, then P ∩ Z 2 is not empty.

This property is trivial due to the fact that we work on the grid Z 2 of size 1.

From a morphological point of view, it can be reformulated as follows.

Property 3 If the erosion of P by a ball

B √ 2 2
of radius

√ 2 2 is not empty, namely, P B √ 2 2 = ∅, then P ∩ Z 2 is not empty. Indeed, if P B √ 2 2 = ∅, then the opening (P B √ 2 2 ) ⊕ B √ 2 2 is the ( √ 2
2 )-regular part of P , noted Reg(P ). Its digitization with grid interval 1 is guaranteed to have the same topological structure of P (see Proposition 1). In this context, the regular part of P is defined by

Reg(P ) = (P B √ 2 2 ) ⊕ B √ 2 2
, and P \ Reg(P ) is called non-regular part. If, for each boundary half-plane of H ∈ R(P ), there exists a ball B √ 2 2 ⊂ P being tangent to H, then we can define the corner parts of P as P \ Reg(P ). More precisely, each connected component of P \ Reg(P ) corresponds to a corner part of P . In other words, the set of the corner parts is defined by Cor(P ) = C(P \ Reg(P )) where C is the (continuous) connected component function (see Fig. 5).

Property 4 Let us consider a corner part A ∈ Cor(P ). If the angle θ(A) at the corresponding vertex of P verifies θ(A) ≥ π

2 and A ∩ Z 2 is not empty, then any point p ∈ A ∩ Z 2 has at least one 4-adjacent point in Reg(P ).

Proof. As the Euclidean distance from p to Reg(P ) cannot be higher than

√ 2
2 , thus inferior to 1, the circle of radius 1 whose center is p always has a non-empty intersection with Reg(P ). Due to the angle condition of θ(A), this intersection forms a circular arc whose central angle is not less than π 2 . The 4-neighbours of p are on the circle of radius 1 and at least one of them appears on this circulararc intersection with Reg(P ). Therefore, there always exists at least one of the 4-neighbours of p in Reg(P ).

2 )-regular (a) and not (b-c): (b) there is a vertex of P with angle < π 2 ; (c) there is an half-plane H ∈ R(P ) where no ball

B √ 2 2 ⊂ P exists such that H is tangent to B √ 2 2
. In the figures, P are in blue,

B √ 2 2 are in red, P B √ 2 2
is in yellow, Reg(P ) are in pink, and Cor(P ) are in green.

Let us now introduce the notion of quasi-regularity. Definition 6. A convex polygon P is quasi-r-regular if ∀H ∈ R(P ), ∃B r ⊂ P such that H is tangent to B r , and if ∀A ∈ Cor(P ), θ(A) ≥ π 2 . From Property 4, we then obtain the following lemma.

Lemma 1. Let X be an H-convex digital object. If Conv(X) is quasi-( √ 2
2 )-regular, then T Conv (X) is 4-connected.

Finally, we obtain the main proposition from this lemma and Property 1.

Proposition 3. Let X be an H-convex digital object. If Conv(X) is quasi-( √ 2
2 )regular, then T Conv (X) is H-convex.

Moreover, the transformed convex polygon P = H∈R(Conv(X))

T(H) generally does not correspond to the convex hull P of T Conv (X), i.e., P = Conv(P ∩ Z 2 ). However, we always have the following inclusion relation. T(H) and P = Conv(T Conv (X)). Then, we always have P ⊆ P .

Rigid motions of a non-convex digital object

In the previous section, we showed how to carry out rigid motions of H-convex digital objects with preservation of their H-convexity under the quasi-( √

2 )regular condition. In this section, we first present a hierarchical representation of a non-convex digital objects based on convex hulls, called a concavity tree [START_REF] Sklansky | Measuring concavity on a rectangular mosaic[END_REF]. Then, we show how to perform rigid motions of a non-convex digital object via this concavity-tree representation.

Concavity tree of a digital object

A concavity tree, initially introduced in [START_REF] Sklansky | Measuring concavity on a rectangular mosaic[END_REF], is a hierarchical representation of the convex and concave parts of the contour of a digital object, with different levels of detail. This tree structure has been used, e.g., for minimum-perimeter polygon computation [START_REF] Klette | Recursive computation of minimum-length polygons[END_REF][START_REF] Sklansky | Measuring concavity on a rectangular mosaic[END_REF], hierarchical shape analysis [START_REF] Borgefors | Analyzing nonconvex 2D and 3D patterns[END_REF], polygonal approximation [START_REF] Aguilera-Aguilera | The computation of polygonal approximations for 2D contours based on a concavity tree[END_REF]. The root of the tree corresponds to a given digital object. Then, each node corresponds to a concavity part of its parent. Note that each concavity part X of a digital object X is obtained as a connected component of the subtraction of X from the digitized convex hull Conv(X) ∩ Z 2 . This is written by X ∈ C((Conv(X) ∩ Z 2 ) \ X) where C(S) denotes the set of all connected components of a finite set S ∈ Z 2 . In other words, we have

X = Conv(X) Z 2 \ X ∈C((Conv(X)∩Z 2 )\X) X , (7)
where each concavity part X is recursively replaced by the subtraction of the concavity parts of X from Conv(X ) Z 2 until no concavity part is found. An illustration of concavity tree is given in Fig. 6.

Digital object rigid motions using concavity tree

As observed in Eq. ( 7), a digital object X can be decomposed into the set of digitized convex hulls of hierarchical concavity parts, Conv(X) ∩ Z 2 , Conv(X ) ∩ Z 2 , . . ., via the subtraction operations. As we are rather interested in such a hierarchical decomposition by digitized convex polygons, we consider a concavity tree such that each node corresponds to the half-plane representation of the convex hull of a concavity part Y, Conv(Y) =

H∈R(Conv(Y))

H, instead of Y itself.

Once the hierarchical object decomposition is obtained, we simply apply T Conv to each digitized convex polygon Conv(Y) ∩ Z 2 , as shown in the previous section, and then carry out the set subtraction operations guided by the tree. Figure 7 illustrates comparisons between applications of point-by-point transformation T point and convexity-preserving transformation T Conv using concavity tree to non-convex digital objects.

From Proposition 3, we can preserve the H-convexity of each digitized convex polygon in the tree if the convex polygon is quasi-

( √ 2
2 )-regular. Practically, such hypothesis is not satisfied in general (see Fig. 6 (b)). However up-sampling strategies on the digital objects can allow us to generate convex polygons which guarantee the quasi-( √

2 )-regularity. This subject is one of our perspectives.

Conclusion

In this article, we proposed a method for rigid motions of digital objects which preserves the H-convexity. The method is based on the half-plane representation of H-convex digital objects. In order to guarantee the H-convexity, we introduced the notion of quasi-r-regularity for convex polygons, and showed that the H-convexity is preserved under rigid motions if the convex hull of an initial H-convex digital object is quasi-

( √ 2
2 )-regular. The necessity of such condition is caused by the fact that the convexity does not imply the connectivity in Z 2 contrary to R 2 . As we need to re-discretize the transformed convex polygon at the end, it is natural to have a similar notion of the r-regularity of continuous objects, which guarantees the topology after its digitization. The method was also extended to non-convex digital objects using the hierarchical object representation.

The proposed method works only with digital objects satisfying the quasi-

( √ 2
2 )-regular condition on the convex hull. In practice, this condition is difficult to obtain, and this would limit the direct use of the proposed method. However, an up-sampling approach would provide a promising strategy to guarantee the quasi-( √

2 )-regularity of the convex hull of any digital object. It is also observed that the quasi-(

√ 2
2 )-regularity is sufficient but not necessary for the H-convexity preservation. It would be one of our perspectives to find such a sufficient and necessary condition. Another perspective would be to extend the method into higher dimensions.

Fig. 1 .Fig. 2 .

 12 Fig. 1. Convexity alterations under rotation: (a) a convex digital object in 2D, and (b) its non-convex transformed image by a rotation; (c) a convex digital object in 3D, and (d) its non-convex transformed image by a rotation.

Fig. 3 .

 3 Fig. 3. Well-composed digital lines (a, c) with different thicknesses, which remain wellcomposed (d) or not (b) after a point-by-point digitized rigid motion Tpoint. In both cases, the convexity of the digital lines is lost by Tpoint.

Fig. 4 .

 4 Fig. 4. Example of the digital half-plane representation of a digital object. Left: Digital object and its convex hull (convex hull vertices are in red). Right: Digital half-planes extracted from this convex hull.

1 gcd

 1 (wx,wy) (-w y , w x ), and c = (a, b) • u where (w x , w y

Fig. 5 .

 5 Fig. 5. Convex polygons P , which are quasi-( √ 2

Property 5

 5 Let X be an H-convex digital object. Let us consider the two convex polygons such that P = H∈R(Conv(X))

Fig. 6 .

 6 Fig. 6. (a) A digital object. (b) The convex and concave parts of (a) with their convex hulls. (c) The concavity tree of (a) where each colored node corresponds to a concavity part of its parent and the root (in orange) corresponds to (a).

Fig. 7 .

 7 Fig. 7. Original digital objects (a,d) with their point-by-point digital rigid motions (b,e) and their convex-preserving digital rigid motions using concavity tree (d,f).

Here, the boundary of S is associated to the continuous boundary induced by the pixels of S, i.e., the union of pixel edges shared by pixels of S and S.

In this paper, we consider only X such that the area of Conv(X) is not null.