Uniform convexity and the splitting problem for selections

TL;DR

This paper explores the splitting problem for continuous set-valued selections in infinite-dimensional uniformly convex Banach spaces, providing new geometric insights and affirmative solutions for specific classes of mappings.

Contribution

It introduces the use of Polyak's notion of uniform convexity and modulus of convexity for convex sets, extending the splitting problem solutions to broader contexts.

Findings

Affirmative solution for the splitting problem in certain uniformly convex set-valued mappings

Development of geometric properties of uniformly convex sets in Banach spaces

Extension of splitting problem results beyond convex balls to general convex sets

Abstract

We continue to investigate cases when the Repov\v{s}-Semenov splitting problem for selections has an affirmative solution for continuous set-valued mappings. We consider the situation in infinite-dimensional uniformly convex Banach spaces. We use the notion of Polyak of uniform convexity and modulus of uniform convexity for arbitrary convex sets (not necessary balls). We study general geometric properties of uniformly convex sets. We also obtain an affirmative solution of the splitting problem for selections of certain set-valued mappings with uniformly convex images.