Hereditary invertible linear surjections and splitting problems for selections

TL;DR

This paper investigates the splitting of continuous mappings into sums of selections from convex sets in Banach spaces, establishing conditions under which such splittings exist, especially focusing on hereditary invertibility of linear surjections.

Contribution

It introduces hereditary invertibility of linear surjections as a key concept for splitting continuous selections, providing new affirmative and negative results in this context.

Findings

Splitting is guaranteed by hereditary invertibility in certain cases.

Positive results for strictly convex finite-dimensional precompact spaces.

Counterexamples show limitations of hereditary invertibility.

Abstract

Let $A+B$ be the pointwise (Minkowski) sum of two convex subsets $A$ and $B$ of a Banach space. Is it true that every continuous mapping $h:X \to A+B$ splits into a sum $h=f+g$ of continuous mappings $f:X \to A$ and $g:X \to B$? We study this question within a wider framework of splitting techniques of continuous selections. Existence of splittings is guaranteed by hereditary invertibility of linear surjections between Banach spaces. Some affirmative and negative results on such invertibility with respect to an appropriate class of convex compacta are presented. As a corollary, a positive answer to the above question is obtained for strictly convex finite-dimensional precompact spaces.