Support, Convexity Conditions and Convex Hypersurfaces in Infinite Dimension

TL;DR

This paper extends convexity theory and studies convex hypersurfaces in infinite-dimensional spaces using non-differential convex analysis, focusing on support, convexity conditions, and Jordan hypersurfaces without assuming smoothness.

Contribution

It generalizes the Convexity Theorem for sets without interior and develops an infinite-dimensional version of Jordan hypersurfaces without relying on smoothness.

Findings

Generalized support for closed sets without interior in infinite dimensions

Extended convexity theorem to non-interior sets in infinite-dimensional spaces

Analyzed decomposition of non-closed, non-pointed cones

Abstract

Working in infinite dimensional linear spaces, we deal with support for closed sets without interior. We generalize the Convexity Theorem for closed sets without interior. Finally we study the infinite dimensional version of Jordan hypersurfaces. Our whole work never assumes smoothness and is based exclusively on non-differential Convex Analysis tools and, in particular, on theory of convex cones. A crucial mathematical tool for our results is obtained solving the decomposition problem for non-closed non pointed cones.