R Azhagarasan

Mapping French -Tamil Transl(oc)ations By

If you say 'Tamil' to a French speaking person today, the first image that comes to his or her mind is that of the Tamil shops in the Faubourg Saint-Denis area in Paris that purvey spices, food items, clothes and fancy jewels, restaurants that serve idlis, dosais, vadais and vegetarian thalis, and outlets that sell music and video CDs at a throw away price. French television reports show images of folkloric Tamil rituals like fire-walking and 'Thaipoosam Cavadee' in the French Caribbean, Réunion and Mauritius islands. But the nitty gritty of Tamil culture .

(Ganapathy-Doré 1993) and its interaction with French culture are matters of concern for public administrators, social workers, the police, judges, translators and specialists (Moudiappanadin 1995). If you say 'French' to a Tamil speaking person, the images that he or she will conjure up are those of revolutionary ideals, artistic and literary avant-gardes, wines and luxury goods.

People who live in Pondicherry might eventually refer to architectural experimentation with Tamil and French elements in Auroville. . Scholars on both sides of the linguistic frontier have tried to go beyond the clichés and get across a more informed view of each other's language and culture.

This article tries to focus on the significant moments in the French-Tamil translations/transactions with a view to highlighting the two following arguments: 1. Unlike the translation activities in the British Indian territories, the translations in the French Indian territories were not strictly guided by the policy of the colonial government and remain voluntary. 2. Since major amount of translation of French to Tamil was done through English, this translation history raises questions relating to the a) nature of choice, b) mode of translation, c) complex relation between the source language (French) and mediating language (English) and the target language (Tamil). Here the 'function' of translation becomes more important than the 'correctness' and 'ethics' of translation. These two arguments taken together hint at how the postcolonial debate on India is most often guided by a critique of British colonialism and fails to capture the nuances of French colonialism.

While using the term 'postcolonial' to refer to the period, the paper distances itself from the idea of 'postcolonial translation' popularized by scholars like Harish Trivedi [START_REF] Bassnett | Postcolonial Translation: Theory and Practice[END_REF] in so far the French to Tamil translations gain significance not for the employment of any strategies, but for their translocations and their function in the Tamil context. Hence the objective of this article is to give an overall picture of the nature and function of translation instead of a micro-analysis. The survey (from colonial to postcolonial period) undertaken here helps us confirm that in terms of quality and quantity, translations from French to Tamil directly or via English are more common than the other way around.

Such an approach helps us suggest that the translations of French texts mediated through English first during the nationalist struggle, then in the hay-days of literary modernism and the rise of Tamil Nationalism and finally during the rise of the feminist and dalit activism had become socially significant and had a much larger impact than the direct French to Tamil translations done with institutional support. The paper also touches upon the role of Tamils outside the Tamil region in this history of transactions between French and Tamil. To accomplish these intertwined arguments the paper follows a chronological pattern, tracing the function of French-Tamil translation from the times of French Orientalism through the rise of nationalism, modernism, and the activist mo(ve)ments of the postcolonial present.

The exchange between French and Tamil cultures was initiated thanks to the French East India Company on whose behalf François Martin bought Pondicherry in 1674. However, the pioneering work in cultural translation came from other Europeans. The first Tamil catechism issued by Portuguese Jesuits at Goa in 1577 was the first work ever to be printed in Indian script. The Bible was first translated into Tamil by Bartholomäus Ziegenbalg, a German Pietist and printed in 1714 in the Danish colony of Tharangambadi [START_REF] Shaw | An initiative that backfired[END_REF]. Interpreters like Veera Naicker (Gobalakichenane 1992) . and Ananda Rangapoullé [START_REF] Vinson | Les Français dans l'Inde, Dupleix et Labourdonnais[END_REF] Foucault, Christian Metz. The context of the 1990s provided a fertile ground for the dissemination of French critical theories. Nevertheless one point to be remembered about these translations is that in their enthusiasm to introduce French authors to Tamil readers, the translators and the journals do not respect copyright regulations. Apart from the transactions of theory, the 90s uprising in India with its focus on minority, gender and caste question was seen on par with the May 68 revolution of France. The comparison was not based on the modus operandi but in terms of its impact on the intellectual and public sphere [START_REF] Tharu | Introduction[END_REF].

From the available texts translated or adapted from French to Tamil which are more numerous than the works translated from Tamil to French, we could discern more curiosity, openness, interest and sustained effort on the part of Tamil authors, readers and publishers to keep abreast of the French literary scene as part of their cosmopolitan culture. This might have been facilitated by traditional forms of cultural exchange such as student exchange, book fairs, tourism and migration. The number of these translated works, however, confirms the colonial pattern of knowledge flow from the West to the East, showing that much remains to be done to right this uneven exchange. The fact that Francophone writing from Africa is less conspicuous in this corpus shows that the gaze is turned towards the West and North (France, Canada) rather than the South.

More than the blind pursuit of the passion for theory in an imitation of the West, the Tamil engagement with French theory shows the Tamil penchant for intellectual debate.

Besides, Dalit thinkers turn to alternative frameworks of thought to change the dominant way of looking and seeing. However, Tamils could be self-deprecatory as is shown by the fictitious interview with Jean-Claude Ivan Yarmola (Oodagam 1994) which laughs at Tamil pseudo intellectuals who run after visiting Western masters without checking their bonafides. . Indeed Alain Sokal's article denouncing the intellectual mumbo-jumbo in France has been translated into Tamil. 3. French publishers complain about the lack of trained translators. They sometimes resort to the easier way out, i.e., back translating Tamil works from their English versions. They are also afraid that the typical concerns of the Tamils may not be attractive to the French market. For some earlier and older migrants from Tamil Nadu, Pondicherry or Sri Lanka, the business of getting ahead in life is so tough that they do not have the leisure time to devote to translation into French of Tamil works by way of leaving a cultural legacy to their children. The Pondicherry families belonging to the 'renonçants'4 category take pride on their assimilation with the French. Their relationship to the French culture somewhat resembles that of the piedsnoirs from Algeria.

A glance at this complex history of French-Tamil translations/transactions also help us look at the nature of French colonization in the Tamil region. Unlike the transactions of British and Russian literatures, French-Tamil transaction is unique. In the case of the British the Colonial administration and Colonial educational policy promoted English literature in Indian languages. In the case of Russian literature, Indo-Russian relationships during the Nehruvian era played a crucial role. But in the case of French literature, there was no official policy of promoting of French literature in India before the politics of francophonie entered into play. The general consensus was that the exclusive use of French was needed only where native languages were too poor in linguistic resources. Otherwise French could coexist alongside native languages. The reception of French works by Tamils was thus voluntary. Indeed French works served as a counterpoint to the dominant English tastes. Hence their long standing impact. That is the reason why critics hesitate to use the anti-colonial rhetoric of postcolonialism while discussing Tamil French transactions. The fact that the French works created such an impact (through English) in both the French and British-Indian territories testify to the unique nature of French-Tamil transactions.

The time is perhaps ripe for a Tamil writer of French expression to emerge. Born to an Indian father and a Tunisian mother, Geetha Balvannanathan Prodhom, is experimenting in this direction by writing poems and short stories directly in French, while continuing to write in Arabic, English and Italian. . The Mauritian writer of Tamil origin Sooriamoorthy Ramanujam is auto-publishing poetry in the French language. . The Sri Lankan community in France is, however, very active in the transmission of Tamil language and culture to the younger generation. Some of its members have established schools to teach Tamil and published bilingual dictionaries and glossaries [START_REF] Sachchithanantham | Dictionnaire pratique français-tamoul-srilankais[END_REF][START_REF] Sachchithanantham | Vocabulaire thématique Français -Tamoul[END_REF] . The Tamil speaking community in Canada seems to be engaged in politics5 as a mode of cultural translation rather than in translation proper of Tamil texts into French. The Tamil diaspora in the French Caribbean, . Reunion Island . and Mauritius [START_REF] Ravi | Rethinking Global Mauritius[END_REF] . are keen to discover their roots. But their knowledge of Tamil is not surefooted enough to undertake ambitious translations of major works. They rely on universities in France and in Tamil Nadu and the cooperation between them to write new chapters in the history of Tamil French connections.

The recent Cannes award to Jacques Audiard's movie Dheepan foregrounded the life of Sri Lankan Tamil refugees in France. . For the first time, the lead roles were played by Tamil actors Jesuthasan Antonythasan, Kalieaswari Srinivasan and Claudine Vinasithamby. In this French movie, Tamil language dialogues were subtitled in French. Indeed Sri Lankan Tamil writing about their displacement is a corpus that needs to be scrutinized in a separate study. Only K.P.

Aravindan's poems have been translated into French by A. Murugaiyan in 2014. . Thanks to globalization, French companies work in Tamil Nadu and Pondicherry where French native speakers face the reality of Tamil language. Though English serves as lingua franca, doing business with Tamil people without showing a respect for their language is not easy in the Tamil context. But the paradox of globalization is such that the Lycée français in Pondicherry is offering courses in English. The overall picture one gets is, thus, one of an immense gap that will take decades to fill. We are not even close to the never accomplished transition that Sandro European powers. We are thus able to witness the continuation of a unique tradition of French-Tamil transaction across times.

  who was the 'Dubash' of Joseph-François Dupleix had facilitated contact between the French and Tamil communities. Several French travellers of the 18 th century went to the Tamil country and Pondicherry and explained Tamil manners and customs to the French public[START_REF] Deleury | Les Indes florissantes, Anthologie des voyageurs français (1750-1820[END_REF]. Before the French Catholic priest Pierre Brigot of the Société des Missions set up a mission in Pondichérry in 1777, Jesuites from Italy like Robert Nobili and Constantine Joseph Beschi had already travelled to the Tamil country and mastered the Tamil language. Beschi (1680-1747) had even composed a Tamil epic called Thembavani on the Life of Saint Joseph. The French missionaries held these works in great esteem. Les Missions étrangères de Paris holds in its archives the original versions of the unsurpassed works Louis Savinien Dupuis and Louis Musset who developed impressive bilingual dictionaries, French grammar manuals for Tamil speakers and bilingual conversation books. . L'imprimerie de la mission in Pondicherry is still the place where French texts are laid out and printed. The Auroville Press set up in 1982 offers services in many languages including French. In the UNESCO's collection of representative works appear only three books translated from Tamil into French (Le roman de l'anneau du Prince Ilango Adigal translated by Alain Danielou in 1990, Le livre de l'amour de Tiruvalluvar translated by François Gros in 1992 and Viramma translated by Jean-Luc and Josiane Racine in 1996) compared to the wider range of books translated from Sanskrit and Hindi into French. . The first two books do not aim at challenging a Sanskrit-centric Orientalism such as what we see in the English translations of Thiruvalluvar by F.W. Ellis (1812) and G.U. Pope (1886). The third text, Viramma (co-authored by Viramma herself) seeks to challenge the strictly limited idea of 'dalit autobiography' circulated through the English translations of dalit texts. The writings of the nationalist Tamil poet Bharati continues to shape the idea of modernism in Tamil. While scholars take pains to explore the nature of his contact with Western literature, particularly French, we do not have clear proof his Tamil translations of French texts. Just because he operated from the French-Indian territory, Pondicherry, and has acquaintance with the French, we cannot assume that he directly translated from the French to Tamil. The same argument applies to the arch-Modernist Pudumai Pithan and the leader of the Dravidian movement, Annadurai. It was only during the period of the little-magazine movement we get ample evidence of French to Tamil translations mediated through English. A brief analysis of the history of French to Tamil translations is given below to highlight the significant moments which facilitated the advent of Nationalism/regionalism and Modernism in Tamil literature. In the early years of the 20 th century, the revolutionary Tamil poet Bharati had exhorted the Tamils to translate treaties written by great minds from other countries into Tamil. He showed the way by translating the first two strophes of the French national anthem into Tamil. . Literary magazines such as Kandeepam published in the 1940s started publishing translations of short stories and novels from other languages. Some Tamil writers started translating and adapting English, French, German and Russian novels into Tamil in the early years of the 20 th century (See Annex. The list heavily relied on Murugesa Pandiyan's 2004 book on World Literature translated into Tamil). That the choice of works translated was mediated by English reading habits is clears from the popularity of authors like Alexandre Dumas and Jules Verne. Their adventure narratives held an appeal for Tamil readers and exercised an influence in the shaping the novel genre in Tamil. Kalki Krishamurthy's novels such as Ponniyin Selvan or Sivagamiyin Sabadham bear the imprint of French narrative techniques. This does not mean that the Tamil people who are known for their lyrical imagination and flair for the drama were not interested in poetry and theatre. But the major interest was for long prose narratives. Most French writers were translated from English either by translators or authors who translated and adapted French works (especially the short stories of Maupassant) by way of learning the trade. Most of them did not come from the French speaking territory of Pondicherry or Karaikal. The titles of the translations, generally, show that the authors sometimes stick to the original, sometimes translate the English title and sometimes 'indigenize' it. The treatment of sexuality by French writers like Balzac, Flaubert, Maupassant and Zola was perceived as less hypocritical by Tamil readers who were used to Victorian prudishness on the one hand and the typical Tamil family loyalty which limited their individual freedom, on the other. The first ever French book to be directly translated from French into Tamil remains, however, Alexandre Dumas's Le Comte de Monte-Cristo by N. C. Gopala Krishnapillai in 1914 under the title Amarasimman. . In the years before independence, French thought struck a chord with the Tamil yearning for dignity and freedom from oppression. Anthologies of world writers and abridged versions of famous novels were also seeing the day. French literature was one of the sources of growing Tamil cosmopolitan consciousness. Special attention was devoted to French Enlightenment philosophers such as Voltaire, Montesquieu and Rousseau. There were also multiple translations of Victor Hugo's Les Misérables or Notre Dame de Paris. They bear witness to the emotional identification of the Tamils with the poor and downtrodden depicted in Victor Hugo's novels. The pathos contained in his novels touched them more profoundly than Charles Dickens's novels. C.N. Anndurai, the founder of the political party Dravida Munnetra Kazhagam (DMK) who was to become the Chief Minister of Tamil Nadu in 1967, published his translation of the first strophe of 'La Marseillaise' as early as 1940. . The 1952 translation of Emile Zola's biography from English to Tamil in two parts by the DMK leader S.P. Sittrarasu in 1952 was followed by C.N. Annadurai's passionate assessment of Zola's work in favour of the poor in 1959 under the title Ezhai Pangalan. DMK journals such as Dravida Nadu, Mandram, Kanchi, Thenral, Ennam continued to disseminate French thought and creative writing. . Several translations of Zola's works were published in the 1960s. In the development and shaping of Tamil rationalist political movements, the influence of Zola and his world vision, if not considerable, was certainly, undeniable. The fact that Vaiko, founder and General Secretary of the Marumalarchi Dravida Munnetra Kazhagam (MDMK) who was held in custody on charges of sedition by the Tamil Nadu police referred to Zola's 'J'accuse' to defend himself shows how ingrained Zola's ideas are in Tamil political consciousness. .1 Tamil publishing industry during the 1960s and 70s largely relied on private enterprise rather than government subsidy and institutional support. In the 1970s, the Tamil monthly Prakgnai published a translation of Camus's Les Justes. The attention shifted to absurdist philosophy thenceforth, prompted perhaps by the disillusionment with postcolonial governance. A stream of books directly translated from French with proper purchase of translation rights followed. V. Sriram, Madanacalliany Shanmuganandam and T.S. Dakshinamurthy are some noteworthy translators. As V. Sriram had also published short essays on French directors like François Truffaut (1997), Robert Cartier Bresson (1998) and Louis Malle (1999), the French government had recognized his services to French culture by giving him a couple of awards, Chevalier dans l'ordre des palmes académiques and chevalier dans l'ordre des arts et letters in 2002. Award winning Tamil Writer and translator Nagarathinam Krishna and his friend and academic Vengada Soupraya Nayagar deserve special mention in this paper for their translations of French and francophone writers into Tamil. Jagadesan Surya's M.Phil thesis submitted in 2015 to Jawarhal Nehru University, A Critical Bibliography of Translations from French into Tamil during 1991-2010, is a welcome addition to Tamil-French cross cultural studies. While the French Embassy and the Alliance Française of Chennai extend their unfailing support to V. Sriram, Indian and Canadian universities seem to have given their encouragement to a group of academics comprising P. Kichenamourty, K. Madanagobalane, R. Venguattaramane and S. Pannirselvame who have translated Tamil short stories into French of course, but more significantly French writers and Francophone Canadian writers into Tamil. Sharan Kumar Subramanian has devoted his doctoral thesis (2014) to a crucial review of these translations of Canadian francophone authors into Tamil. Contemporary Tamil critics have turned to French theory as a way of maintaining a distance from the dogmas and orthodoxies of dominant Indian culture. How and why did they turn what has been termed as 'French theory' by François Cusset . (Cusset 2005) is an interesting issue. In order to answer this question, one has to find out which French theories have been translated and how these theories have travelled to the Tamil publishing world. Tamil journals published in Tamil Nadu such as Niraprigai, Ayvu, Padigal and Sidhaivu explore French theories, while Uyirnizhal, a journal published by Sri Lankan Tamils led by Pradeepan Ravindran in France pays attention to French critical thought as well as literary and artistic creations and experimentations. Sometimes main stream journals like Droit et cultures publish in -depth articles about Tamil literature (Annousamy 2007). . Apart from Pierre Bourdieu's Sur la télévision (On Television) translated by Sriram in 2004, no French theoretician has been directly translated from French to English. The translations have been made from their English translations. The English translations of French theories were prescribed as part of research methodology in University curricula. On the one hand, the well-known Indian postcolonialist Gayatri Spivak was also the translator of Derrida might explain the importance given to French theories in Indian academia. It is an irony that despite being a translator of French theory, and an observer of the French trading post in Bengal, Chandranagore, Spivak failed to look into this unique nature of French colonial history. Introduction of French theoreticians started in the late 80s in Madras University and Pondicherry University. The fact that Roland Barthes's 'Death of the Author,' Michel Foucualt's Archeology of Knowledge and Jacques Lacan's 'Of Structure as an Inmixing of an Otherness Prerequiste to any Subject Whatever' feature in the M.A. English syllabus for the academic year (2008-2009) of Thiruvalluvar University 2 in Serkadu . is just an example of how French literary theories have penetrated the Indian academic world. Mother Teresa Women's University in Kodaikanal made it a point to introduce French feminist theories to its students. R. Azhagarasan's Tamil translation (2009) of Poststructuralism by Catherine Belsey . is an example of mediation between Tamil and French critical worlds via English. His book is part of the series of translations of Oxford conceptual/critical thinking by Adyalam Publishers. Translated books on Postmodernism by authors Prem (who along with Ramesh claim themselves to be the Deleuze and Guattari of the Tamil Little-magazine circle), Postcolonialism by Mangai and Critical Theory by Sivakumar belong to the same series. Besides, the little magazines, especially those devoted to Dalit causes, were those that continued to offer French critical thought with a view to uplifting the minds of Tamil readers through English. There is a twofold purpose in the recourse to French theories by such magazines. These translations exercised a leveling influence between those who went to University and those who educate themselves by reading Tamil magazines. The most important translators of French critical theories into Tamil from English in the small magazines are D. Ravikumar, Nagarjunan and Sivakumar. The French theoreticians who have been translated into Tamil are: Antonin Arthaud, George Bataille, Albert Camus, Jean-Paul Sartre, Aimé Césaire, Jean-Baudrillard, Gilles Deleuze, Claude Levi-Strauss, Michel

http://mdmk.org.in/article/mar09/i-accuse, published by Vaiko on 4 July

Accessed on July 25, 2015.

http://thiruvalluvaruniversity.ac.in/pdf/cbcs_syllabus/M.A.%20English.pdf. Accessed July 25, 2015.

Posted by Stefan Steinberg in the World Socialist Web Site on 1 st July 2000..

According to the Décret du 18 septembre 1881 of the Third Republic, the natives could change their status from subject to citizens and choose a patronym for themselves and their heirs.

Rathika Sitsabaiesan is the first person of Tamil origin to be elected to the Canadian parliament.

Mezzadra and Federico Rahola . identify as defining the postcolonial condition (Mezzadra and Rahola, 2006).

Among the actors of cultural translation, literary journals and websites as well as associations need to be mentioned. The French language journals published in India such as Rencontre avec l'Inde (published by the Indian Council for Cultural Relations founded in 1950), Synergies Inde (since 2006), but also Courrier de l'Unesco, . Nouvelles de l'Inde published by the Embassy of India in France, La Nouvelle revue de l'Inde, Indes -La fenêtre surl'Inde, sur le monde francophone as well as scholarly journals such as Hommes et migrations and Puruṣ ārtha, disseminate information about Tamil language and culture (but not exclusively) among the In the present context of globalization, the nature of transaction between French and Tamil goes beyond translation of texts. Here too the nature French engagement with globalization and the Tamil diaspora seems to maintain a difference, when compared to other


Mohammed Bachir
email: mohammed.bachir@univ-paris1.fr

ON THE KREIN-MILMAN-KY FAN THEOREM FOR CONVEX COMPACT METRIZABLE SETS

Keywords: Extreme points, exposed and Φ-exposed points, Φ-convexity, Krein-Milman theorem, variational principle. 2010 Mathematics Subject: 46B22, 46B20, 49J50

The Krein-Milman theorem (1940) states that every convex compact subset of a Hausdorff locally convex topological space, is the closed convex hull of its extreme points. In 1963, Ky Fan extended the Krein-Milman theorem to the general framework of Φ-convexity. Under general conditions on the class of functions Φ, the Krein-Milman-Ky Fan theorem asserts then, that every compact Φ-convex subset of a Hausdorff space, is the Φ-convex hull of its Φextremal points. We prove in this paper that, in the metrizable case the situation is rather better. Indeed, we can replace the set of Φ-extremal points by the smaller subset of Φ-exposed points. We establish under general conditions on the class of functions Φ, that every Φ-convex compact metrizable subset of a Hausdorff space, is the Φ-convex hull of its Φ-exposed points. As a consequence we obtain that each convex weak compact metrizable (resp. convex weak * compact metrizable) subset of a Banach space (resp. of a dual Banach space), is the closed convex hull of its exposed points (resp. the weak * closed convex hull of its weak * exposed points). This result fails in general for compact Φ-convex subsets that are not metrizable.

where ϕ i ∈ Φ and λ i ∈ R for all i ∈ I. For a nonempty set A ⊆ S, the intersection of all Φ-convex subset of S contaning A is said to be the Φ-convex hull of A. By conv Φ (A) we denote the Φ-convex hull of A. Let a, x, y ∈ S, we say that a is Φ-between x and y, if (ϕ ∈ Φ, ϕ(x) ≤ ϕ(a), ϕ(y) ≤ ϕ(a)) =⇒ (ϕ(a) = ϕ(x) = ϕ(y)).

Let ∅ = A ⊆ B ⊆ S. The set A is said to be Φ-extremal subset of B, if (a ∈ A, a is Φ-between the points x, y ∈ B) =⇒ (x ∈ A, y ∈ A).

If A is a singleton A = {a}, we say that a is Φ-extremal point of B. The set of all Φ-extremal points of a nonempty set A will be denoted by ΦExt(A).

When S is a Hausdorff locally convex topological vector space (in short l.c.t space, "Hausdorff" will be implicit), and Φ = S * is the topological dual of S, then the Φ-convexity and the classical convexity coincides for closed subsets of S and the Φ-convex hull of a set coincides with its closed convex hull. Also, the Φ-extremal points of a nonempty set, coincides with its extreme points [ [START_REF] Khanh | Sur la Φ-Convexité de Ky Fan[END_REF], Proposition 2.] ( Note that in [START_REF] Khanh | Sur la Φ-Convexité de Ky Fan[END_REF], the author use a class Ψ of functions in the definitions which correspond to the class -Φ in the above definitions. The class Φ concidered in this paper will be a vector space). Recall that if C is a subset of S, we say that a point x ∈ C is an extreme point of C, and write x ∈ Ext(C), if and only if :

y, z ∈ C, 0 < α < 1; x = αy + (1 -α)z =⇒ x = y = z.
The result in what is known as the Krein-Milman theorem (1940, [START_REF] Krein | On extreme points of regular convex sets[END_REF]), asserts that if K is a convex compact subset of an l.c.t space, then K is the closed convex hull of its extreme points,

K = conv(Ext(K)).
The Krein-Milman theorem has a partial converse known as Milman's theorem (See [START_REF] Phelps | Lectures on Choquet's Theorem[END_REF]) which states that if A is a subset of K and the closed convex hull of A is all of K, then every extreme point of K belongs to the closure of A,

(A ⊂ K; K = conv(A)) =⇒ Ext(K) ⊂ A.
In [START_REF] Fan | On the Krein-Milman theorem[END_REF], Ky Fan extended the Krein-Milman theorem to the more general framework of Φconvexity.

Theorem 1. (Krein-Millman-Ky Fan) Let S be a Hausdorff space and Φ a familly of real valued functions defined on S. Let K be a nonempty compact Φ-convex subset of S and suppose that:

(1) the restriction of each ϕ ∈ Φ to K, is lower semicontinuous on K;

(2) Φ separate the points of K. Then, ΦExt(K) = ∅ and K = conv Φ (ΦExt(K)).

The main results of the paper. Recall that when S is an l.c.t space and C is a subset of S, we say that a point x ∈ C is an exposed point of C, and write x ∈ Exp(C), if there exists some continuous linear functional x * ∈ S * which attains its strict maximum over C at x. Such a functional is then said to expose C at x. Thus, an exposed point is a special sort of extreme point. If S is a dual space, a weak * exposed point x (we write x ∈ w * Exp(C)) is to simply an exposed point by a continuous functional from the predual. We introduce a general concept of Φ-exposed points that coincides with the classical exposed points when S is an l.c.t space and Φ = S * and coincides with the weak * exposed points where S = E * is a dual space and Φ = E. Definition 1. Let S be a Hausdorff space, C a nonempty subset of S and Φ a familly of real valued functions defined on S. We say that a point x of C is Φ-exposed in C, and write x ∈ ΦExp(C), if there exists ϕ ∈ Φ such that ϕ has a strict maximum on C at x i.e. ϕ(x) > ϕ(y) for all y ∈ C \ {x}. Such ϕ is then said to Φ-expose C at x.

It is easy to see that ΦExp(C) ⊆ ΦExt(C) but the converse is not true in general. The first main result of this paper is the following theorem.

Theorem 2. Let S be a Hausdorff space and (Φ, . Φ ) be a Banach space of real valued functions defined on S. Let K be a nonempty compact metrizable Φ-convex subset of S and suppose that:

(1) the restriction of each ϕ ∈ Φ to K is continuous, and there exists some real number α K ≥ 0 such that α K ϕ Φ ≥ sup x∈K |ϕ(x)| for all ϕ ∈ Φ;

(2) Φ separate the points of K. Then, we have that (i) ΦExp(K) = ∅ and the set of all ϕ ∈ Φ that Φ-expose K at some point, contains a dense G δ subset of (Φ, . Φ );

(ii) K = conv Φ (ΦExp(K)).

We give below some examples in the linear framework, where the above theorem can be applied (for more details see the corollaries of Subsection 3.1).

Examples 1. Let (E, . ) be a Banach space, S = (E, Weak) and (Φ, . Φ ) = (E * , . ) the topological dual of (E, . ). In this case, a subset of S is Φ-convex iff it is convex and weak closed iff it is convex and norm closed (by Mazur's lemma on the coincidence of weak and norm closure of convex sets). The Φ-convex hull of a set coincides with its weak closed convex hull which also coincides with its norm closed convex hull (by Mazur's lemma). The Φ-exposed points coincides with the classical exposed points.

Examples 2. Let (E, . ) be a Banach space, S = (E * , Weak * ) and let (Φ, . Φ ) = (E, . ). In this case, a subset of S is Φ-convex iff it is convex weak * closed and the Φ-convex hull of a set, coincides with its weak * closed convex hull. The Φ-exposed points coincides with the weak * exposed points.

Note that Theorem 2 fails in general for compact subsets that are not metrizable. For example, when (E, . ) = (l 1 (Γ), . 1 ) (Γ uncountable set), S = (E * , Weak * ) and (Φ, . Φ ) = (E, . ), then, all of the hypothesis of Theorem 2 are satisfed. However, for the not metrizable weak

* compact subset K = B E * , we have that ΦExp(K) = w * Exp(K) = ∅ (See Remark 4).
The second main result of this paper is the following theorem. The space (C(K), . ∞ ) denotes the Banach space of all real valued continuous functions on K equiped with the supnorm. Let (Φ, . ∞ ) be a closed Banach subspace of (C(K), . ∞ ). By B Φ * we denote the dual unit ball of (Φ, . ∞ ). We also use the following notation:

±δ(ΦExp(K)) := {±δ k /k ∈ ΦExp(K)}
where, for each k ∈ ΦExp(K), δ k : ϕ → ϕ(k) for all ϕ ∈ Φ. Theorem 3. Let K be a compact metric space and (Φ, . ∞ ) be a closed Banach subspace of (C(K), . ∞ ) which separates the points of K and contains the constants. Then, we have

w * Exp(B Φ * ) = ±δ(ΦExp(K)),
and

B Φ * = conv w * (±δ(ΦExp(K))).
Note that in Theorem 3, the concept of Φ-exposed points of K appears in a natural way in the description of the weak * exposed points of B Φ * . As consequence, we deduce, under the hypothesis of Theorem 3, that the set ΦExp(K) of all Φ-exposed points of K, is a dense subset of the Shilov boundary ∂Φ of Φ i.e. ∂Φ = ΦExp(K) (Corollary 8).

The results of this paper are based on the following version of variational principle in the compact metric framework (Lemma 3 in Section 2). This analogous of the Deville-Godefroy-Zizler variational principle [START_REF] Deville | A smooth variational principle with applications to Hamilton-Jacobi equations in infinite dimensions[END_REF], also gives a new information about the set of "ill-posed problems" on compact metric sets. Several others consequences are obtained (the details are given in Section 2).

A key lemma. Let K be a compact metric space and (Y, . Y ) be a Banach space included in C(K) which separates the points of K and such that α . Y ≥ . ∞ for some real number α ≥ 0. Let f : K → R ∪ {+∞} be a proper lower semi-continuous function. Then, the set In a separable Banach space Y , each set N which can be covered by countably many dc hypersurf ace is σ-lower porous, also σ-directionally porous; in particular it is both Aronszajn (equivalent to Gauss) null and Γ-null. For more details about "small sets", see [START_REF] Zajicek | On sigma-porous sets in abstract spaces[END_REF] and references therein. This paper is organized as follows. In Section 2, we prove the key lemma (Lemma 3) and give several consequences. In Section 3 we give the proofs of the main results (Theorem 2 and Theorem 3) and applications. In Section 4 we give some additional properties and remarks about Φ-exposed points and the Krein-Milman theorem.

N (f ) = {ϕ ∈ Y : f -ϕ does not

Variational principle and consequences.

Let (M, d) be a complete metric space and f : M -→ R ∪ {+∞} be an extended real-valued function which is bounded from below and proper. By the term proper we mean that the domain of f , dom(f

) := {x ∈ M/f (x) < +∞} is non-empty. We say that f has a strong minimum at x if inf M f = f (x) and d(x n , x) → 0 whenever f (x n ) → f (x).
The problem to find a strong minimum for f , is called Tykhonov well-posed-problem. Let (C b (M ), . ∞ ) be the space of all real-valued bounded and continuous functions on M , equipped with the sup-norm and let (Y, . Y ) be a Banach space included in C b (M ). Let

N (f ) = {ϕ ∈ Y : f -ϕ does not have a strong minimum on M } .
The set N (f ) is called the set of "ill-posed problems". The problem is to find conditions on Y under which the set N (f ) is a "small" set. In [START_REF] Deville | A smooth variational principle with applications to Hamilton-Jacobi equations in infinite dimensions[END_REF], Deville, Godefroy and Zizler proved that the set N (f ) is of first Baire category in Y , and in [START_REF] Deville | Porosity of ill-posed problems[END_REF], Deville and Rivalski generalize the result of Deville-Godefroy-Zizler (D-G-Z), where they showed that the set N (f ) is σ-porous in Y, whenever f is bounded from below, proper and lower semi continuous and Y satisfies the following conditions:

(i) g ≥ g ∞ , for all g ∈ Y ;

(ii) for every natural number n, there exists a positive constant M n such that for any point x ∈ M there exists a function

h n : M -→ [0; 1], such that h n ∈ Y , h n ≤ M n , h n (x) = 1 and diam(supp(h)) < 1 n .
The D-G-Z variational principle has several applications in particular in optimization and in the geometry of Banach spaces and can be applied without compactness assumption. However, the assumption (ii) in the above result is crutial and so the D-G-Z variational principle cannot includes the linear case, like the Stegall's variational principle. Of course, the interest in the D-G-Z variational principle, is to circumvent the compactness, but in our purpose ( in connection with the Krein-Milman theorem), we need only to treat the compact framework. Thus, we prove in the key Lemma 3 that when we assume that (K, d) is compact metric space, the condition (ii) can be omitted. This allows to expand the class Y to a class of functions including the linear cases. Moreover, if (Y, . Y ) is a separable Banach space included in C(K) and separate the points of K, then the set N (f ) can be more smaller than σ-porous. In fact we prove that in this situation, the set N (f ) can be covered by countably many d.c. hypersurf ace (See the definitions below). This gives, in particular, examples showing that the σ-porousity of the "ill-posed problems" in [START_REF] Deville | Porosity of ill-posed problems[END_REF], is not optimal. Our version of variational principle has several consequences, in particular it allows to give the proofs of the main results of the paper. The proof of Lemma 3, is based on the use of a differentiability result of convex continuous functions on a separable Banach spaces due to Zajicek [START_REF] Zajicek | On the differentiation of convex functions in finite and infinite dimensional spaces[END_REF] and a non convex analogue to Fenchel duality introduced in [START_REF] Bachir | A non convexe analogue to Fenchel duality[END_REF].

We recall from [START_REF] Zajicek | On sigma-porous sets in abstract spaces[END_REF] the following definitions.

Definition 2. Let Y , Z be Banach spaces, C ⊂ Y an open convex set, and F : C → Z a continuous mapping. We say that F is d.c. (that is, delta-convex) if there exists a continuous convex function f :

C → R such that y * • F + f is convex whenever y * ∈ Y * , y * ≤ 1.
Definition 3. Let Y be a Banach space and n ∈ N * , 1 ≤ n < dimY . We say that A ⊂ X is a d.c. surf ace of codimension n if there exist an n-dimensional linear space F ⊂ X, its topological complement E and a d.c. mapping ϕ : E → F such that A = {x + ϕ(x) : x ∈ E}. A d.c. surface of codimension 1 will be called a d.c. hypersurf ace.

2.1. Preliminary lemmas. If (Y, . Y ) is a Banach space included in C b (M ) with α . ≥ . ∞
for some real number α ≥ 0 and x ∈ M , we denote by δ x the evaluation map (Dirac mass) on Y at x i.e. δ x : ϕ -→ ϕ(x), for all ϕ ∈ Y . The map δ x is a linear continuous functional on Y since α . ≥ . ∞ . We recall the following definition from [START_REF] Bachir | A non convexe analogue to Fenchel duality[END_REF]. Definition 4. Let (M, d) be a complete metric space and (Y, . Y ) be a Banach space included in C b (M ) with α . ≥ . ∞ for some real number α ≥ 0. We say that the space Y has the property P G if and only if, for every sequence (x n ) n ⊂ M , the following assertions are equivalent: (i) the sequence (x n ) n converges in (M, d), (ii) the associated sequence of the Dirac masses (δ xn ) n converges in (Y * , Weak * ).

The letter G in P G is justified by the fact that the Gâteaux bornology, the Gâteaux differentiability and the weak * topology has some connection between them. We refer to [START_REF] Bachir | A non convexe analogue to Fenchel duality[END_REF] for more details. The space C b (M ), the subspace C u b (M ) of uniformly continuous functions and the space Lip b (M ) of all bounded and Lipschitz continuous functions (equipped with their natural norms), satisfies the property P G for any complete metric space (M, d) (see [Proposition 2.6, [START_REF] Bachir | A non convexe analogue to Fenchel duality[END_REF]]). The following Lemma shows that, in the compact metric framework, the property P G is satisfied for a large class of function spaces.

Lemma 1. Let (K, d) be a compact metric space and (Y, . Y ) be a Banach space included in C(K), which separates the points of K and such that α . ≥ . ∞ for some real number α ≥ 0. Then, Y has the property P G .

Proof. If (x n ) n is a sequence of K that converges to some point x in (K, d), it is clear that (δ xn ) n converge to δ x for the weak * topology. Suppose now that (δ xn ) n converge to some point Q in Y * for the weak * topology. We prove that the sequence (x n ) n converge in (K, d). Indeed, suppose that l 1 and l 2 are two distinct cluster point of (x n ) n . There exists two subsequences

(y n ) n and (z n ) n such that (y n ) n converge to l 1 and (z n ) n converge to l 2 . Since (δ xn ) n converge to Q and (Y * , Weak * ) is a Hausdorff space, it follows that δ l1 = Q = δ l2
which is a contradiction since Y separate the points of K. So, the sequence (x n ) n has a unique cluster point, and hence it converges to some point since K is a compact metric space. Now, if we are interested on the property P G for separable Banach spaces (Y, . Y ) included in C b (M ), the following proposition shows that, this situation holds only when M is compact. In fact, this characterizes the compact metric sets. Proposition 1. Let (K, d) be a complete metric space and (Y, . Y ) be a separable Banach space included in C(K), which separate the points of K and such that α . ≥ . ∞ for some real number α ≥ 0. Then, the following assertions are equivalent.

(1) K is compact.

(2) Y has the property P G .

Proof. The part (1) =⇒ ( 2) is given by Lemma 1. Let us prove the part (2) =⇒ [START_REF] Araujo | On Shilov boundaries for subspaces of continuous functions[END_REF]. Indeed, Since Y is separable, by the Banach-Alaoglu theorem, the dual unit ball B Y * is a compact metrizable space. Let us denotes δ(K) := {δ k : k ∈ K} and consider the map:

δ : (K, d) → (δ(K), Weak * ) x → δ x
Since Y has the property P G , it follows that (δ(K), Weak * ) is a closed subspace of the compact metrizable set (B Y * , Weak * ). Therefore, (δ(K), Weak * ) is a Hausdorff compact space. Since Y separate the points of K, the map δ is one-to-one. Consequently, δ : (K, d) → (δ(X), Weak * ) is a continuous and bijective map from (K, d) onto the compact space (δ(K), Weak * ), it is then an homeomorphism which implies that (K, d) is a compact space.

We also need the following lemma.

Lemma 2. Let K be a compact metric space and (Y, . Y ) be a Banach space included in C(K). Suppose that Y separate the points of K, then there exists a sequence (ϕ n ) n ⊂ Y which separates the points of K, in particular, there exists a separable Banach subspace Z of (Y, . Y ) which also separates the points of K.

Proof. Since K is a compact metric space, then (C(K), . ∞ ) is a separable Banach space. Since Y is a subspace of C(K), it is also . ∞ -separable. Thus, there exists a sequence (ϕ n ) n ⊆ Y which is dense in Y for the norm . ∞ . Since Y separate the points of K, if x, y ∈ K are such that x = y, then there exists ϕ ∈ Y such that ϕ(x) = ϕ(y).
Using the uniform convergence of a subsequence of (ϕ n ) n to ϕ, we get that the sequence (ϕ n ) n also separate the points of K.

Now, if we set Z = span . Y {ϕ n : n ∈ N}, then Z is a separable Banach subspace of (Y, . Y )
that separates the points of K.

2.2.

The key lemma. Recall that in [START_REF] Zajicek | On the differentiation of convex functions in finite and infinite dimensional spaces[END_REF], Zajicek proved that in a separable Banach space, the set N G(F ) of the points where a convex continuous function F is not Gâteaux differentiable, can be covered by countably many d.c. hypersurf ace. This result together with a duality result from citeBa, will be used in the proof of Lemma 3.

Lemma 3. Let K be a compact metric space and (Y, . Y ) be a Banach space included in C(K) which separates the points of K and such that α . Y ≥ . ∞ for some real number α ≥ 0. Let f : K → R ∪ {+∞} be a proper lower semi-continuous function. Then, the set

N (f ) = {ϕ ∈ Y : f -ϕ
∈ Z by f × (ϕ) := sup x∈K {ϕ(x) -f (x)}. It is clear that f × is a convex 1-Lipschitz continuous function on Z.
The separable case: Suppose that Z is a separable Banach subspace of (Y, . Y ) which separates the points of K. Using [Theorem 2; [START_REF] Zajicek | On the differentiation of convex functions in finite and infinite dimensional spaces[END_REF]] we get that f × is Gâteaux-differentiable outside a set N which can be covered by countably many d.c hypersurf ace in Z. On the other hand, combining Lemma 1 and [Theorem 2.8., [START_REF] Bachir | A non convexe analogue to Fenchel duality[END_REF]] we get that f × is Gâteaux-differentiable at a point ϕ ∈ Z if and only if fϕ has a strong minimum on K. Thus, the set N of points of non Gâteaux-differentiability of f × , coincide with the set

N (f ) ∩ Z = {ϕ ∈ Z : f -ϕ does not have a strong minimum on K} .
This complete the proof in the separable case.

The general case: As in the proof of Deville-Godefroy-Zizler variational principle, we use the Baire category theorem, but here we dont admits the existence of a bump function in Y having a support of arbitrary small diameter. We use The separable case. Indeed, we claim that

O n := {ϕ ∈ Y ; ∃x n ∈ K/(f -ϕ)(x n ) < inf{(f -ϕ)(x) : d(x, x n ) ≥ 1 n }}
is an open dense subset of Y . The fact that O n is open follows from the hypothesis α . Y ≥ . ∞ . Let us prove that O n is dense. From Lemma 2 there exists a separable Banach subspace Z of (Y, . Y ) which separates the points of K. Let ϕ ∈ Y and ε > 0. By using "The separable case" with the function fϕ and the separable space Z, we get an h ∈ Z such that h Y < ε and (fϕ)h has a strong minimum on K at some point x 0 . It follows that ϕ + h ∈ O n for all n ∈ N * , by taking x n = x 0 for all n ∈ N * . This shows that O n is dense in Y .

Since Y is a complete metric space, G = ∩ n≥1 O n is a dense G δ subset of Y , by the Baire category theorem. Now, we claim that

G ⊆ {ϕ ∈ Y : f -ϕ have a strong minimum on K} . Indeed, let ϕ ∈ G. For each n ≥ 1 there exists x n ∈ K such that (f -ϕ)(x n ) < inf{(f -ϕ)(x) : d(x, x n ) > 1 n }.
Since K is compact metric space, there exists a subsequence (x n k ) k that converges to some point x ∞ . Using the lower semicontinuity of f , we get that

(f -ϕ)(x ∞ ) ≤ lim inf k (f -ϕ)(x n k ) ≤ lim inf k inf{(f -ϕ)(x) : d(x, x n k ) > 1 n k } ≤ inf{(f -ϕ)(x) : x ∈ K \ {x ∞ }} Now, to see that x ∞ is a strong minimum of f -ϕ, let (y n ) n be a sequence in K such that (f -ϕ)(y n ) converges to (f -ϕ)(x ∞ ).
We prove that (y n ) n converges to x ∞ . Indeed, suppose that the contrary hold. Extracting, if necessary, a subsequence, we can assume that there exists ε > 0 such that for all n ∈ N , d(y n , x ∞ ) ≥ ε. Thus, there exists an integer p such that d(x p , y n ) ≥ 1 p for all n ∈ N. Hence,

(f -ϕ)(x ∞ ) ≤ (f -ϕ)(x p ) < inf{(f -ϕ)(x) : d(x, x p ) ≥ 1 p } ≤ (f -ϕ)(y n )
for all n ∈ N, which contedict the fact that lim n (fϕ)(y n ) = (fϕ)(x ∞ ). This concludes the proof.

We obtain immediately the following Corollary.

Corollary 1. Let (K, d) be a compact metric space and Y be any closed subspace of (C(K), . ∞ ) that separates the points of K. Let f : K → R ∪ {+∞} be a proper lower semi-continuous function. Then, the set Proof. Since (K, d) is a compact metric space, (C(K), . ∞ ) is a separable Banach space and so (Y, . ∞ ) is a separable Banach subspace satisfying the hypothesis of Lemma 3.

N (f ) = {ϕ ∈ Y : f -ϕ
The above Corollary cannot be obtained from the D-G-Z variational principle. For example, the space Y = {ϕ ∈ C(B R n )/ϕ(0) = 0}, where B R n denotes the closed unit ball of R n for a fixed norm, satisfies the hypothesis of Corollary 1 but does not satisfy the condition (ii) of Deville-Rivalski:

(ii) for every natural number n, there exists a positive constant M n such that for any point x ∈ B R n there exists a function

h n : B R n -→ [0; 1], such that h n ∈ Y , h n ≤ M n , h n (x) = 1 and diam(supp(h)) < 1 n .
Remark 1. A strong and strict minimum coincides for lower semi continuous functions on a compact metric space.

2.3.

Applications to the linear variational principle and to the Gâteaux differentiability. As consequence of Lemma 3, we have the following linear analogous to the Stegall's variational principle. Note that also this results cannot be obtained from the D-G-Z variational principle.

Proposition 2. Let E be a Banach space and K be a weak * compact metrizable subset of E * . Let f : (K, Weak * ) -→ R ∪ {+∞} be a proper lower semi-continuous function. Then, the set Recall that a weak Asplund space E is a Banach space in which every convex continuous function is Gâteaux differentiable at each point of a dense G δ subset of E. The following corollary, gives a class of convex continuous functions which are Gâteaux differentiable at each point of a dense G δ subset of E, where E is any Banach space.

N (f ) = {x ∈ E : f -x
Corollary 2. Let E be a Banach space and f : E -→ R, be a convex continuous function such that dom(f * ) Let C be a non-empty subset of E * . We denote by σ C the support function defined on E by σ C (x) = sup

Weak * is weak * compact metrizable subset of E * . Then, f is Gâteaux differentiable at each point of a dense G δ subset of E. Proof. Set K = dom(f * ) Weak * , which is a convex weak * compact metrizable subset of E * . Since, f * : K -→ R ∪ {+∞}
x * ∈C

x * (x); ∀x ∈ E.

Let f : E -→ R ∪ {+∞} be a proper lower semi contiuous convex function. The inf convolution of f and σ C is defined by

f ▽ σ C (x) := inf y∈E {f (x -y) + σ C (y)}.
From the above corollary we get that if K is a convex weak * compact metrizable subset of E * and f is proper lower semi contiuous convex function on E, then f ▽σ K is Gâteaux differentiable at each point of a dense G δ subset of E.

Let K be a convex subset of a vector space. A function ϕ : K → R is said to be affine if for all x, y ∈ K and 0 ≤ λ ≤ 1, ϕ(λx + (1λ)y) = λϕ(x) + (1λ)ϕ(y). The space of all continuous real-valued affine functions on K will be denoted by Af f (K). Proposition 4. Let K be a compact metrizable convex subset of a l.c.t space X and f : K -→ R ∪ {+∞} be a proper lower semi-continuous function. Then the set N (f ) := {ϕ ∈ Af f (K) : fϕ does not have a strong minimum on K} can be covered by countably many d.c hypersurface of (Af f (K), . ∞ ).

Proof. We use Lemma 3 with Y = Af f (K). Since (Af f (K), . ∞ ) is a closed Banach subspace of the separable Banach space (C(K), . ∞ ), it is separable. On the other hand, by the Hahn-Banach theorem, Af f (K) separate the points of K, since it contains the set {x * |K : x * ∈ X * }. So, from Lemma 3, the set N (f ) = {ϕ ∈ Af f (K) : fϕ does not have a strong minimum on K} can be covered by countably many d.c hypersurface of (Af f (K), . ∞ ).

The main results and applications.

This section is devoted to the proofs of the main results of the paper. Some applications are also given.

3.1. Proof of Theorem 2 and consequences. Now, we give the proof of the first main result.

Proof of Theorem 2. Let K be a compact metrizable Φ-convex subset of S. Under the hypothesis, replacing if necessary (Φ, . Φ ) by the Banach space (Y, . Y ) where Y := {ϕ |K , the restriction of ϕ ∈ Φ to K} with ϕ |K Y := ϕ Φ , we can assume that (Φ, . Φ ) is a Banach space included in C(K) which separates the points of K and satisfies α . Φ ≥ . ∞ . Thus, Lemma 3 applies. Using Lemma 3, applied with the lower semi continuous function f = i K (the indicator function which is equal to 0 on K and +∞ othewize), we get that the set of all ϕ ∈ Φ which Φ-exposes K at some point, contains a dense G δ subset of Φ. In particular ΦExp(K) = ∅.

Since K is Φ-convex subset of S, it is clear that conv Φ (ΦExp(K)) ⊆ K. Now, let us prove that K = conv Φ (ΦExp(K)). Suppose towards a contradiction that there exists k 0 ∈ K \ conv Φ (ΦExp(K)).

Claim. There exist h ∈ Φ and r ∈ R such that

sup{h(k) : k ∈ conv Φ (ΦExp(K))} < r < h(k 0 ).
Proof of the claim. Since conv Φ (ΦExp(K)) is a Φ-convex subset of S, then there exists a set

I, ϕ i ∈ Φ and λ i ∈ R for all i ∈ I such that conv Φ (ΦExp(K)) = i∈I {k ∈ S/ϕ i (k) ≤ λ i }.
Since k 0 ∈ conv Φ (ΦExp(K)), there exits i 0 ∈ I such that ϕ i0 (k 0 ) > λ i0 . On the other hand, we have ϕ i0 (k) ≤ λ i0 for all k ∈ conv Φ (ΦExp(K)). This finish the proof of the claim by taking h = ϕ i0 and by chosing a real number r ∈ R such that λ i0 < r < ϕ i0 (k 0 ). Now, using Lemma 3 applied with f = -h + i K , we can find ψ ∈ Φ close to 0 in Φ such that h + ψ, Φ-expose K at some point k 1 ∈ ΦExp(K). Since α . ≥ . ∞ , we have that ϕ := h + ψ is also close to h uniformly on K. Hence, using the claim, ϕ satisfies also

sup{ϕ(k) : k ∈ conv Φ (ΦExp(K))} < r < ϕ(k 0 ). (1)
On the other hand

ϕ(k 0 ) ≤ sup{ϕ(k) : k ∈ K} = ϕ(k 1 )
which is a contradiction with (1), since k 1 ∈ ΦExp(K).

For the classical convexity we obtain the following Krein-Milman type results for convex compact metrizable subsets, where the extreme points are replaced by the exposed points. Note that the part (2) in the following corollary is an extension of a Klee result [Theorem 2.1, [START_REF] Klee | Extremal structure of convex sets II[END_REF]] (See also [Theorem 4.5, [START_REF] Klee | Extremal structure of convex sets II[END_REF]]) and the part (1) sems to be new in a general Banach space. We know from [Theorem 6.2., [START_REF] Phelps | Convex Functions, Monotone Operators and Differentiability[END_REF]] that a Banach space is a Gâteaux differentiability space if and only if, every convex weak * compact subset of E * is the weak * closed convex hull of its weak * exposed points. Hence, for a non Gâteaux differentiability space (for example if E = l 1 (Γ), Γ uncountable set) there always exist a convex weak * compact subset of E * which is not the weak * closed convex hull of its weak * exposed points. This shows in particular that Theorem 2 is not true in general for not metrizable convex compact subsets. However, the part (1) of the following corollary, shows that the situation is better for convex weak * compact metrizable subset of E * , when E is any Banach space. Corollary 3. Let E be a Banach space.

(1) Let K be a convex weak * compact metrizable subset of E * . Then,

K = conv w * (w * Exp(K)).
(2) Let K be a convex weak compact metrizable subset of E. Then,

K = conv w (Exp(K)) = conv . (Exp(K)).
Proof. In the part (1), we apply Theorem 2 with the convex weak * metrizable subset K of E * and by taking (Φ, . Φ ) = (E, . ). In the part (2), we apply Theorem 2 with the convex weak metrizable subset K of E and by taking (Φ, . Φ ) = (E * , . ), using in this case the fact that the weak and norm closur coincides for convex sets by the well known Mazur's lemma.

In [START_REF] Klee | Extremal structure of convex sets II[END_REF], Klee pointed the fact that outside the normed space, the above result is not true. He suspected that some condition rather close to normability may be needed and that the metrizability is inadequate even in the separable case, mentioning the following counterexample: in the locally convex separable metrizable space R ℵ0 , the cube [-1, 1] ℵ0 has no exposed points. To answer positively this problem in the general l.c.t spaces, we introduce an intermediate concept of remarkable points called "affine exposed points" which is between the concept of exposed points and extreme points. Definition 5. Let K be a convex subset of a l.c.t space X. We say that a point x ∈ K is an affine exposed point of K, and write x ∈ AExp(K), if there exists some affine continuous map τ ∈ Af f (K) which attains its strict maximum over K at x.

Clearly, Exp(K) ⊆ AExp(K) ⊆ Ext(K), but these inclusions are strict in general. For example, the cube [-1, 1] ℵ0 has affine exposed points by Proposition 4, but is without exposed points. A comparison of these three sets will be given in Subsection 4.1.

We obtain then the following result.

Corollary 4. Let K be a convex compact metrizable subset of a l.c.t space X. Then, AExp(K) = ∅ and K is the closed convex hull of its affine exposed points

K = conv(AExp(K)).
Proof. The proof is given by taking S = K and (Φ, . Φ ) = (Af f (K), . ∞ ) in Theorem 2.

We also have the following consequences.

Corollary 5. Let E be a Banach space.

(1) Let (K, Weak * ) be a convex weak * compact metrizable subset of E * . Then, the set w * Exp(K) is weak * dense in the set AExp(K), which is weak * dense in the set Ext(K).

(2) Let (K, Weak) be a convex weak compact metrizable subset of E. Then, the set Exp(K) is weak dense in the set AExp(K), which is weak dense in the set Ext(K).

Proof. First, note that the spaces (E * , Weak * ) and (E, Weak) are l.c.t spaces. Combining the part (1) (resp. the part (2)) of Corollary 3 with Corollary 4 and the partial converse of the Krein-Milman theorem, we get the part (1) (resp. the part (2)).

3.2.

Proof of Theorem 3 and the Shilov boundary. In this subsection, we give the proof of the second main result. We need the following lemma from [START_REF] Bachir | An extension of the Banach-Stone theorem[END_REF].

Lemma 4. (See [START_REF] Bachir | An extension of the Banach-Stone theorem[END_REF]) Let Z be a Banach space and h, k : Z → R be two continuous and convex functions. Suppose that the function z → l(z) := max(h(z), k(z)) is Fréchet (respectively, Gâteaux) differentiable at some point z 0 ∈ Z. Then either h or k (maybe both h and k) is Fréchet (respectively, Gâteaux) differentiable at z 0 and l

′ (z 0 ) = h ′ (z 0 ) or l ′ (z 0 ) = k ′ (z 0 ).
Proof. We give the proof for the Fréchet differentiability, the Gâteaux differentiability is similar. Suppose without loss of generality that l(z 0 ) = h(z 0 ) and let us prove that h is Fréchet differentiable at z 0 and that l ′ (z 0 ) = h ′ (z 0 ). For each z = 0 we have:

0 ≤ h(z 0 + z) + h(z 0 -z) -2h(z 0 ) z ≤ l(z 0 + z) + l(z 0 -z) -2l(z 0 ) z .
Since l is convex and Fréchet differentiable at z 0 , then the right-hand side in the above inequalities, tends to 0 when z tends to 0. This implies that h is Fréchet differentiable at z 0 by the convexity of h. Now, if we denote f = hl, then f (z 0 ) = 0, f ≤ 0 and f ′ (z 0 ) exists. Thus, for all z ∈ Z f ′ (z 0 )(z) = lim

t-→0 + 1 t (f (z 0 + tz) -f (z 0 )) ≤ 0.
This implies that f ′ (z 0 ) = 0. Thus h ′ (z 0 ) = l ′ (z 0 ).

We also need to establish the following lemma.

Lemma 5. Let K be a compact metric set and (Φ, . ∞ ) be a closed Banach subspace of (C(K), . ∞ ) which separates the points of K and contains the constants. Then, the following assertions are equivalente.

(

) A point Q ∈ B Φ * is a weak * exposed point (2) there exists a Φ-exposed point k ∈ ΦExp(K) such that Q = ±δ k , where δ k : ϕ → ϕ(k) for all ϕ ∈ Φ. Proof. (1) =⇒ (2). Let Q ∈ w * Exp(B Φ * ) 1
, so there exists ϕ ∈ Φ which weak * expose B Φ * at Q. It follows from [Proposition 6.9., [START_REF] Phelps | Convex Functions, Monotone Operators and Differentiability[END_REF]] that the norm . ∞ is Gâteaux differentiable at ϕ with Gâteaux derivative equal to Q. On the other hand it is clear that ψ ∞ = max(0 × (ψ), 0 × (-ψ)) for all ψ ∈ Φ, where 0 × (ψ) = sup k∈K ϕ(k) for all ϕ ∈ Φ. Thus, from Lemma 4 we have that either ψ → 0 × (ψ) or ψ → 0 × (-ψ) is Gâteaux differentiable at ϕ with Gâteaux derivative equal to Q. Suppose in the first case that is the function ψ → 0 × (ψ) which is Gâteaux differentiable at ϕ. Thus, from Lemma 1 and [Theorem 2.8 [START_REF] Bachir | A non convexe analogue to Fenchel duality[END_REF]] applied with the function f = 0, we get that there exists k ∈ K such that ϕ has a strong maximum at k and that Q = δ k . Thus, in this case k is Φ-exposed by ϕ and Q = δ k . For the second case, where it is the function ψ → 0 × (-ψ) which is Gâteaux differentiable at ϕ with Gâteaux derivative equal to Q, in a similar way, using Lemma 1, [Theorem 2.8 [START_REF] Bachir | A non convexe analogue to Fenchel duality[END_REF]] and the chain rule formula we obtain that there exists some k ∈ K such that -ϕ has a strong maximum at k (so that k is Φ-exposed point) and Q = -δ k .

(2) =⇒ [START_REF] Araujo | On Shilov boundaries for subspaces of continuous functions[END_REF]. Suppose that k ∈ ΦExp(K). There exists ϕ ∈ Φ which Φ-exposes k. Thus -ϕ has a strict minimum at k, equivalent to a strong minimum at k, since K is compact metric set. We can find a real number r such that -(ϕ + r) has also a strong minimum at k and such that ϕ + r > 1 on K. Hence, the function 0 × coincides with . ∞ on an open neighborhood of ϕ + r ∈ Φ. Since -(ϕ + r) has a strong minimum at k, [Theorem 2.8 [START_REF] Bachir | A non convexe analogue to Fenchel duality[END_REF]] asserts that 0 × and so also . ∞ is Gâteaux differentiable at ϕ + r with Gâteaux derivative equal to δ k . It follows from [Proposition 6.9., [START_REF] Phelps | Convex Functions, Monotone Operators and Differentiability[END_REF]], that δ k is weak * exposed by ϕ + r. Thus δ k ∈ w * Exp(B Φ * ). By the symmetry of B Φ * , we also have that -δ k ∈ w * Exp(B Φ * ).

A. The second main result. Now, we give the proof of the second main result.

Proof of Theorem 3. The first part is given by Lemma 5. Now, since (Φ, . ∞ ) is separable, the weak * compact set (B Φ * , Weak * ) is metrizable. Thus, from Corollary 3 applied to the convex compact metrizable set (B Φ * , Weak * ), we have that

B Φ * = conv w * (w * Exp(B Φ * )).
Thus, we have that

B Φ * = conv w * (w * Exp(B Φ * )) = conv w * (±δ(ΦExp(K))).
This concludes the proof.

We deduce immediately the following corollaries. Replacing Φ by Af f (K) in Theorem 3, we obtain: Corollary 6. Let K be a compact metrizable convex subset of a l.c.t space X. Then,

w * Exp(B (Af f (K)) * ) = ±δ(AExp(K)), and
B (Af f (K)) * = conv w * (±δ(AExp(K))), where ±δ(AExp(K)) := {±δ k /k ∈ AExp(K)}.
Replacing Φ by C(K) in Theorem 3, where (K, d) is a compact metric space, and observing that ΦExp(K) = K, since each point k ∈ K is an exposed point by the continuous function x → -d(x, k), we obtain: Corollary 7. Let (K, d) be a compact metric space. Then,

w * Exp(B (C(K)) * ) = ±δ(K), and
B (C(K)) * = conv w * (±δ(K)).
B. The Shilov boundary and the Φ-exposed points. Let K be a compact space and (Φ, . ∞ ) be a closed Banach subspace of (C(K), . ∞ ) which separates the points of K. A subset L of K is said to be a norming subset for Φ if for every ϕ ∈ Φ, we have

ϕ ∞ = sup x∈L |ϕ(x)|. A closed subset C of K is a norming subset for Φ if and only if C is a boundary for Φ, that is, for every ϕ ∈ Φ, we have ϕ ∞ = max x∈C |ϕ(x)|.
The Choquet boundary of Φ, denoted Ch(Φ), is defined as the set of all x ∈ K such that δ x is an extreme point of the unit ball of Φ * . It is well known that Ch(Φ) is a boundary for Φ.

(See [ [START_REF] Taylor | Introduction to Functional Analysis[END_REF], p. 184]). When Φ admits a unique minimal closed boundary, it is called the Shilov boundary of Φ and is denoted by ∂Φ. D. P. Milman proved the existence of Silov boundary for every closed linear subspace of C(K), separating points of K and containing the constants (See [START_REF] Milman | Characteristic of extremal functions[END_REF] and [START_REF] Milman | On integral representations of functions of several variables[END_REF]). He also proved that in this case the Silov boundary concides with the closure of the Choquet boundary: ∂Φ = Ch(Φ). A proof of this result, due to H.S. Bear can be found in [START_REF] Bear | The Silov boundary for a linear space of continuous functions[END_REF]. For other informations about boundary sets we refer to [START_REF] Araujo | On Shilov boundaries for subspaces of continuous functions[END_REF] and [START_REF] Araujo | Linear isometries between subspaces of continuous functions[END_REF]. As a consequence of Theorem 3, we prove below that if K is a compact metric space, then the set of Φ-exposed points ΦExp(K) is a norming subset for Φ and that, its closure coincides with the Shilov boundary of Φ. Note that, ΦExp(K) ⊆ Ch(Φ), but this inclusion is strict in general.

Corollary 8. Let K be a compact metric set and (Φ, . ∞ ) be a closed Banach subspace of (C(K), . ∞ ) which separates the points of K and contains the constants. Then, the set ΦExp(K) is a norming subset for Φ and we have that

∂Φ = ΦExp(K) = Ch(Φ).
Proof. By the Hahn-Banach theorem, for each ϕ ∈ Φ, we have

ϕ ∞ = sup Q∈B Φ * Q, ϕ .
By Theorem 3, we have that

ϕ ∞ = sup Q∈conv w * (±δ(ΦExp(K))) Q, ϕ .
Since the map φ : Q → Q, ϕ is linear and weak * continuous, we obtain that

ϕ ∞ = sup Q∈±δ(ΦExp(K)) Q, ϕ = sup k∈ΦExp(K) |ϕ(k)|.
Thus, the set ΦExp(K) is a norming subset for Y . It follows that ΦExp(K) is a boundary for Φ. It is clear that ∂Φ ⊆ ΦExp(K), since ∂Φ is the minimal closed boundary. We prove that ΦExp(K) ⊆ ∂Φ. Suppose that the contrary hold, there exists k 0 ∈ ΦExp(K) such that k 0 ∈ ∂Φ. Thus there exists ϕ ∈ Φ that expose K at k 0 i.e. ϕ(k 0 ) > ϕ(k) for all k ∈ K \ {k 0 }. On the other hand, since ∂Φ is compact, there exists k 1 ∈ ∂Φ such that ϕ(k 1 ) = sup k∈∂Φ ϕ(k). Thus ϕ(k 0 ) > ϕ(k 1 ) = sup k∈∂Φ ϕ(k). Since Φ contain the constant, there exists r ∈ R such that ϕ + r ∈ Φ and ϕ + r ≥ 0. Hence, we have that

ϕ + r ∞ = ϕ(k 0 ) + r > ϕ(k 1 ) + r = sup k∈∂Φ (ϕ(k) + r) = ϕ + r ∞
which is a contradiction. Thus, ΦExp(K) ⊆ ∂Φ and so we have ΦExp(K) = ∂Φ.

Apendix.

In this section, we give some additional properties about remarkable points and the Kreinmillman theorem.

4.1.

Comparison between exposed, affine exposed and extreme points. Examples. Let K be a convex subset of a l.c.t space X. It is easy to see that we always have

Exp(K) ⊆ AExp(K) ⊆ Ext(K).
This section is devoted to give examples showing that these inclusions are strict in general.

A) Example where Exp(K) AExp(K). The cube [-1, 1] ℵ0 in the locally convex separable metrizable space R ℵ0 , has no exposed points however the set of its affine exposed points is nonempty. Indeed, for example the point b = (1, 1, 1, ...) is affine exposed in [-1, 1] ℵ0 by the affine continuous map defined on [-1, 1] ℵ0 by ϕ : (x 1 , x 2 , x 3 , ...) → n≥0 2 -n x n .

A slight change of the set [-1, 1] ℵ0 , gives also an example where ∅ = Exp(K) = AExp(K). For example we can take the convex compact set

K := {ta + (1 -t)k/t ∈ [0, 1], k ∈ [-1, 1] ℵ0 },
where a = (-2, 0, 0, 0, ...). In this case the point a is exposed by the continuous functional x * : (x 1 , x 2 , x 3 , ...) → -x 1 , but the point b = (1, 1, 1, ...) is not an exposed point. However, b is affine exposed by the affine continuous map defined on K by ϕ : (x 1 , x 2 , x 3 , ...) → n≥0 2 -n x n .

B) Example where Exp(K) = AExp(K). Let K be a convex compact subset of an l.c.t space. Clearly, all translates of continuous linear functionals are elements of Aff(K), but the converse in not true in general (see the above example. See also [START_REF] Phelps | Lectures on Choquet's Theorem[END_REF] page 22.). However, we do have the following relationship. If (E, . ) is a Banach space and E * is its topological dual, the space X = (E * , Weak * ) is a l.c.t space. It is well know that in this case we have that X * = E (See for instance [Corollary 224., [START_REF] Habala | Introduction to Banach Spaces[END_REF]]). In this case, the exposed points of a subset of X coincides, by definition, with the weak * exposed points and the closure of a subset coincides with the weak * closure. Proposition 6. Let E be a Banach space. Let K be a convex weak * compact subset of E * such that the norm interior of K is nonempty. Let X be the l.c.t space (E * , Weak * ). Then, X * = E and w * Exp(K) = AExp(K).

Proof. The fact that X * = E, follows from [Corollary 224., [START_REF] Habala | Introduction to Banach Spaces[END_REF]]. Now, let a be a point in the interior of K. Replacing K by Ka we can assume without loss of generality that 0 belongs to the interior of K. Thus, from [Corollary 224., [START_REF] Habala | Introduction to Banach Spaces[END_REF]], each linear functional that is continuous on K, belongs to the space E. This shows that

{x * |K + r : x * ∈ X * , r ∈ R} = {x |K + r :
x ∈ E, r ∈ R} where x denotes the map x * → x * (x) for all x * ∈ E * . It is easy to see that the space {x |K + r : x ∈ E, r ∈ R} equipped with the sup-norm on K is isomorphic to (E ⊕ R, . + |.|), since K is norm bounded and contain 0 in its (norm) interior. Hence, {x * |K +r : x * ∈ X * , r ∈ R} is a closed Banach subspace of (Af f (K), . ∞ ). This implies by Proposition 5 that Af f (K) = {x |K + r : x ∈ E, r ∈ R}.

Note that since X * = E, by definition we have that the weak * exposed points of the set K considered as a subset of the dual Banach space (E * , . ) coincides with the exposed points of K considered as a subset of the l.c.t space X = (E * , Weak * ). Note also that a map x|K + r affine expose K if and only if x|K expose K. Hence, w * Exp(K) = AExp(K). Proposition 7. In normed vector space, the exposed points and the affine exposed points coincides for a nonempty convex compact set of finite-dimension (i.e compact convex set whose affine hull is finite-dimensional).

Proof. Let V be a normed vector space and K be a convex compact set of finite-dimension. Up to a translation, we can assume without loss of generality that 0 ∈ K. Let V 0 be the linear hull of K. Then K has nonempty interior in V 0 , since V 0 is finite-dimentional. Hence, we have that Af f (K) = {x * |K + r/x * ∈ V * 0 ; r ∈ R} by Proposition 6 (since, weak and norm topology coincides in finite-dimentional). Thus, if ϕ ∈ Af f (K), affine expose k ∈ K, then there exists x * ∈ V * 0 and r ∈ R such that x * |K + r expose K at k. This is equivalent to the fact that x * expose K at k (since r is a constant). Now, by the Hahn-Banach theorem, there exists x * ∈ V * such that x * coincides with x * on V 0 . Hence, x * ∈ V * also expose K at k. This shows that k is an exposed point of K.

Remark 3. We know from Proposition 5 that the set

L(K) := a ∈ Af f (K) : a = r + x * |K f or some x * ∈ X * and some r ∈ R is dense in (Af f (K), . ∞ ).
As it is given in Proposition 6, there exists situations where the sets L(K) and Af f (K) coincides, for instance if K = B E * in (E * , Weak * ), where E is a Banach space. There exist also other situations, where L(K) can be very "small" subset of Af f (K). Indeed, if K is a compact metrizable subset of a l.c.t space X, without exposed points (for example if K = [-1, 1] ℵ0 in R ℵ0 ), then from Proposition 4, we get that L(K) ⊆ N (0) and so L(K) can be covered by countably many d.c hypersurface in (Af f (K), . ∞ ).

C) Example where AExp(K) Ext(K). It is well known that even in the two dimensional space R 2 , there exists a closed unit ball B for a suitable norm, such that Exp(B) = Ext(B) (See for instance Examples 5.9 in [START_REF] Phelps | Convex Functions, Monotone Operators and Differentiability[END_REF]). Thus by Proposition 6 we have also that AExp(B) = Ext(B).

Remark 4. Note that the Corollary 3 and Corollary 4 fails for convex compact sets which are not metrizable. Indeed, take K = B E * where E = l 1 (Γ) (Γ is uncountable set), we know that the norm . 1 is nowhere Gâteaux differentiable (See Example 1.4 (b) p. 3 in [START_REF] Phelps | Convex Functions, Monotone Operators and Differentiability[END_REF]). So from [Proposition 6.9., [START_REF] Phelps | Convex Functions, Monotone Operators and Differentiability[END_REF]] we get that the dual unit ball B (l 1 (Γ)) * in the l.c.t space ((l 1 (Γ)) * , Weak * ), has no (weak * ) exposed points. It follows from Proposition 6 that w * Exp(B (l 1 (Γ)) * ) = AExp(B (l 1 (Γ)) * ) = ∅. Note also that the assumption of local convexity cannot be omitted. Indeed, Roberts, proved in [20] (1977), that there exist a Hausdorff topological vector space X which is metrizable by a complete metric, and a nonempty compact convex set K ⊆ X such that Ext(K) = ∅.

4.2.

Remarks on the A.E.P.P spaces. We introduce the following class of l.c.t spaces. Definition 6. An l.c.t space X is said to have the "Affine Exposed Points Property" (in short A.E.P.P.) if and only if every convex compact subset of X is the closed convex hull of its affine exposed points.

Let us define

Ξ := {X l.c.t space in which every compact subset is metrizable}.

The class Ξ, has been actively studied in the 80 ′ s years by several authors. This class contains of course all metrizable l.c.t spaces, in particular Fréchet spaces but is much larger. For several examples, we refer to [START_REF] Cascales | On Compactness in Locally Convex Spaces[END_REF] and references therein.

We obtain immediately from Corollary 4 the following corollary. Corollary 9. Every space from the class Ξ, has the A.E.P.P.

In particular the space R ℵ0 has the A.E.P.P. Examples of l.c.t spaces having the A.E.P.P. who do not belong to the class Ξ are given in Remark 5. For an example of an l.c.t space without A.E.P.P. we mention the l.c.t space ((l 1 (Γ)) * , Weak * ), where Γ is uncountable (See Remark 5 below). Thus, spaces having A.E.P.P. encompasses a broad class of spaces and it would be interesting to better know their properties.

Examples 3. Immediate examples.

(1) Every Fréchet space has the A.E.P.P.

(2) Every convex closed and bounded subset of a Fréchet-Montel space is the closed convex hull of its affine exposed points (in Fréchet-Montel space, any closed bounded set is compact metrizable). A classical example of a Fréchet-Montel space is the space C ∞ (Ω) of smooth functions on an open set Ω in R n .

Recall that a Banach space E is said to be a Gâteaux differentiability space (GDS) iff each convex continuous real valued function defined on E is Gâteaux differentiable at each point of a dense subset. In [START_REF] Phelps | Convex Functions, Monotone Operators and Differentiability[END_REF], Phelps proved the following result.

Theorem 4. ([Theorem 6.2., [START_REF] Phelps | Convex Functions, Monotone Operators and Differentiability[END_REF] p. 95]) A Banach space E is a GDS if and only if every weak * compact convex subset of E * is the weak * closed convex hull of its weak * exposed points.

Remark 5. (1) Since the exposed points are in particular affine exposed points, it follows from the above theorem that the space (E * , Weak * ) has the A.E.P.P. whenever E is a GDS. However, if E is a non separable GDS, the dual unit ball is a weak * compact not metrizable subset. Thus, the space (E * , Weak * ) has the A.E.P.P. but (E * , Weak * ) / ∈ Ξ, whenever E is a nonseparable GDS (For example the nonseparable Hilbert spaces).

(2) The l.c.t space ((l 1 (Γ)) * , Weak * ) does not have the A.E.P.P. (See Remark 4). More generally, the l.c.t space (E * , Weak * ) does not have the A.E.P.P. whenever E is a Banach space equipped with a nowhere Gâteaux differentiable norm.

For examples of not metrizable spaces which belongs to the class Ξ, we have for example: Proposition 8. Let E be a separable Banach space. Then (E * , Weak * ) and (E, Weak) belongs to the class Ξ, in particular, they have the A.E.P.P. but are not metrizable.

Proof. It is well known that the whole spaces (E * , Weak * ) and (E, Weak) are not metrizable. It is also well known that a Banach space E is separable iff every compact subset of (E * , Weak * ) is metrizable. Thus, (E * , Weak * ) ∈ Ξ. For the space (E, Weak), let K be a weak compact subset of E. Since E is separable, then K is also separable. Now, consider K as a subset of E * * by the canonical embedding, we get that K is norm separable and weak * compact subset of E * * , which implies from [Lemma 2, [START_REF] Bachir | Limited operators and differentiability[END_REF]] that K is weak * metrizable in E * * . In other words, K is weak metrizable. Thus (E, Weak) ∈ Ξ.

Several others not trivial examples of spaces from Ξ can be found in [START_REF] Cascales | On Compactness in Locally Convex Spaces[END_REF].

  have a strict minimum on K} is of the first Baire category in Y . If moreover Y is separable then N (f ) can be covered by countably many d.c. hypersurface in Y .

  does not have a strong minimum on K} can be covered by countably many d.c. hypersurface in Y .

  is proper weak * lower semi continuous, by Proposition 2 we have that the set G = {x ∈ E/f *x has a strict minimum on K} contain a G δ subset of E. Since by definition, dom(f * ) ⊂ K, we have that G = {x ∈ E/f *x has a strict minimum on E * }. By using the classical Asplund-Rockafellar duality result [Corollary 1, [3]], we get that f is Gâteaux differentiable at each point of G.

Proposition 5 .

 5 ([19], Proposition 4.5) Assume that K is a compact convex subset of an l.c.t space X, thenL(K) := a ∈ Af f (K) : a = x *|K + r f or some x * ∈ X * and some r ∈ R is dense in (Af f (K), . ∞ ).

  does not have a strict minimum on K} is included in a set of the first Baire category. If moreover, E is a separable Banach space, then N (f ) can be covered by countably many d.c. hypersurface in E.Proof. Since K is weak * compact in E * , it is norm bounded. Let α := sup x * ∈K x * < +∞. Thus, (E, . ) is a Banach space, included in C(K), which separates the points of K and satisfies α . ≥ . ∞ . So we can apply Lemma 3 with (Y, . ) = (E, . ) to obtain the result. Proposition 3. Let E be a Banach space. Let K be a weak compact metrizable subset of E.

Let f : (K, Weak) -→ R ∪ {+∞} be a proper lower semi-continuous function. Then, the set

N (f ) = {x * ∈ E * : fx * does not have a strict minimum on K} is included in a set of the first Baire category in E * . If moreover, E * is a separable Banach space, then N (f ) can be covered by countably many d.c. hypersurface in E * . Proof. Since K is weak compact in E * , it is norm bounded. Let α := sup x∈K x < +∞. Thus, (E * , .

) is a Banach space, included in C(K), which separates the points of K by the Hahn-Banach theorem and satisfies α . ≥ . ∞ . So we can apply Lemma 3 with (Y, . ) = (E * , . ) to obtain the result. Remark 2. From the above propositions, we get in particular that the set of all continuous functionals from the predual that not exposes the dual unit ball B E * of a separable Banach space E, can be covered by countably many d.c. hypersurface in E. Also in a reflexive separable Banach space E, the set of all continuous functionals that not exposes the unit ball B E can be covered by countably many d.c. hypersurface in E * .