On the splitting problem for selections

TL;DR

This paper explores conditions under which the splitting problem for selections has solutions in finite-dimensional Banach spaces, identifying specific geometric conditions that guarantee affirmative answers and providing counterexamples.

Contribution

It establishes that the splitting problem has positive solutions when images are P-sets or strictly convex, and solves the approximate splitting problem in Hilbert spaces.

Findings

Splitting problem solvable when images are P-sets or strictly convex

Counterexample showing no solution in D space for simple cases

Affirmative solution for approximate splitting in Hilbert spaces

Abstract

We investigate when does the Repov\v{s}-Semenov Splitting problem for selections have an affirmative solution for continuous set-valued mappings in finite-dimensional Banach spaces. We prove that this happens when images of set-valued mappings or even their graphs are P-sets (in the sense of Balashov) or strictly convex sets. We also consider an example which shows that there is no affirmative solution of this problem even in the simplest case in $\R^{3}$. We also obtain affirmative solution of the Approximate splitting problem for Lipschitz continuous selections in the Hilbert space.