Extending the Choquet theory: Trace convexity

Mohammed Bachir (1) , Aris Daniilidis (2, 3)

1 SAMM - Statistique, Analyse et Modélisation Multidisciplinaire (SAmos-Marin Mersenne)
2 UAB - Departament de Matemàtiques [Barcelona]
3 DIM - Departamento de Ingeniería Matemática [Santiago]

We introduce the notion of trace convexity for functions and respectively, for subsets of a compact topological space. This notion generalizes both classical convexity of vector spaces, as well as Choquet convexity for compact metric spaces. We provide new notions of trace-convexification for sets and functions as well as a general version of Krein-Milman theorem. We show that the class of upper semicontinuous convex-trace functions attaining their maximum at exactly one Choquet-boundary point is residual and we obtain several enhanced versions of the maximum principle which generalize both the classical Bauer's theorem as well as its abstract version in the Choquet theory. We illustrate our notions and results with concrete examples of three different types.

Mots clés: Choquet convexity; Exposed point; Krein-Milman theorem; Trace-convexity; Bauer Maximum Principle; Variational Principle; Extreme point.

Domaines: Analyse fonctionnelle [math.FA]

Mohammed Bachir, Aris Daniilidis. Extending the Choquet theory: Trace convexity. 2020. ⟨hal-02531787⟩