Local characterization of strongly convex sets

TL;DR

This paper investigates local properties of strongly convex sets in Hilbert spaces, introducing a generalized modulus of convexity and analyzing its limit behavior as a key characterization tool.

Contribution

It introduces a generalized modulus of convexity for strongly convex sets and establishes the existence of its limit ratio as epsilon approaches zero.

Findings

The generalized modulus of convexity effectively characterizes strongly convex sets.

The limit of the ratio of the modulus to epsilon squared exists for closed convex sets.

Local properties can fully describe strong convexity in Hilbert spaces.

Abstract

Strongly convex sets in Hilbert spaces are characterized by local properties. One quantity which is used for this purpose is a generalization of the modulus of convexity \delta_\Omega of a set \Omega. We also show that \lim_{\epsilon \to 0} \delta_\Omega(\epsilon)/\epsilon^2 exists whenever \Omega is closed and convex.