Convexity of the images of small balls through perturbed convex multifunctions

TL;DR

This paper proves that certain convex multifunctions maintain the convexity of small balls under perturbations in Banach spaces, extending the Polyak convexity principle and impacting set-valued optimization theory.

Contribution

It extends the Polyak convexity principle to perturbed convex multifunctions in Banach spaces, showing preservation of convexity of small balls under perturbations.

Findings

Convexity of small balls is preserved under perturbations of convex multifunctions.

The result applies to mappings in uniformly convex Banach spaces.

Applications to set-valued optimization are discussed.

Abstract

In the present paper, the following convexity principle is proved: any closed convex multifunction, which is metrically regular in a certain uniform sense near a given point, carries small balls centered at that point to convex sets, even if it is perturbed by adding C^{1,1} smooth mappings with controlled Lipschizian behaviour. This result, which is valid for mappings defined on a subclass of uniformly convex Banach spaces, can be regarded as a set-valued generalization of the Polyak convexity principle. The latter, indeed, can be derived as a special case of the former. Such an extension of that principle enables one to build large classes of nonconvex multifunctions preserving the convexity of small balls. Some applications of this phenomenon to the theory of set-valued optimization are proposed and discussed.