L'analyse idempotente etudie les espaces lineaires de dimension infinie dans lesquels l'operation maximum se substitue a l'addition habituelle. Nous demontrons un ensemble de resultats dans ce cadre, en soulignant l'interet des outils d'approximation fournis par la theorie des domaines et des treillis continus. Deux champs d'etude sont consideres : l'integration et la convexite. En integration idempotente, les proprietes des mesures maxitives a valeurs dans un domaine, telles que la regularite�� au sens topologique, sont revues et completees ; nous elaborons une reciproque au theoreme de Radon-Nikodym idempotent ; avec la generalisation Z de la theorie des domaines nous depassons differents travaux lies aux representations de type Riesz des formes lineaires continues sur un module idempotent. En convexite tropicale, nous obtenons un theoreme de type Krein-Milman dans differentes structures algebriques ordonnees, dont les semitreillis et les modules idempotents topologiques localement convexes ; pour cette derniere structure nous prouvons un theoreme de representation integrale de type Choquet : tout element d'un compact convexe K peut etre represente par une mesure de possibilite supportee par les points extremes de K. Des reflexions sont finalement abordees sur l'unification de l'analyse classique et de l'analyse idempotente. La principale piste envisagee vient de la notion de semigroupe inverse, qui generalise de facon satisfaisante a la fois les groupes et les semitreillis. Dans cette perspective nous examinons les proprietes "miroir" entre semigroupes inverses et semitreillis, dont la continuite fait partie. Nous elargissons ce point de vue en conclusion

https://pastel.archives-ouvertes.fr/pastel-00666633/document

Infinite-dimensional idempotent analysis : the role of continuous posets

We show analogues of the classical Krein-Milman theorem for several ordered algebraic structures, especially in a semilattice (non-linear) framework. In that case, subsemilattices are seen as convex subsets, and for our proofs we use arguments from continuous lattice theory and abstract convexity theory.

Convexities on ordered structures have their Krein--Milman theorem

On Some Infinite Convex Invariants

This is a survey on support and separation properties of convex sets in the n-dimensional Euclidean space. It contains a detailed account of existing results, given either chronologically or in related groups, and exhibits them in a uniform way, including terminology and notation. We first discuss classical Minkowski's theorems on support and separation of convex bodies, and next describe various generalizations of these results to the case of arbitrary convex sets, which concern bounding and asymptotic hyperplanes, and various types of separation by hyperplanes, slabs, and complementary convex sets.

/pdf/support-and-separation-properties-of-convex-sets-infinite-1awb92pfz2.pdf

Support and separation properties of convex sets infinite dimension

© Foundation Compositio Mathematica, 1977, tous droits réservés. L’accès aux archives de la revue « Compositio Mathematica » (http: //http://www.compositio.nl/) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.

Some intersection and generation properties of convex sets

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Some intersection and generation properties of convex sets