New Characterizations of Strong Convexity

TL;DR

This paper introduces new ways to characterize strong convexity, focusing on sets that are intersections of closed balls of equal radius, expanding understanding beyond existing equivalences.

Contribution

It provides novel characterizations of r-strong convexity, linking it to intersections of closed balls, complementing previous results on prox-regularity and convexity.

Findings

New characterizations of r-strong convexity

Sets can be expressed as intersections of equal-radius closed balls

Extends understanding of convexity properties in geometric terms

Abstract

Parallel to the main results of [13] and [14], which explore the equivalence between prox-regularity, the exterior sphere condition, and $S$-convexity, we present novel characterizations of the $r$-strong convexity property, namely, of the sets that can be expressed as the intersection of closed balls with the same radius $r>0$.