On Strongly Convex Sets and Farthest Distance Functions

TL;DR

This paper introduces a polarity concept for sets in Banach spaces, characterizes strongly convex sets via farthest distance functions, and explores their properties and relationships.

Contribution

It defines a new polarity notion for sets in Banach spaces and characterizes strongly convex sets through their farthest distance functions.

Findings

Second polar of a set equals the smallest strongly convex set containing it

Farthest distance functions characterize strongly convex sets

Properties of farthest distance functions related to strong convexity

Abstract

A polarity notion for sets in a Banach space is introduced in such a way that the second polar of a set coincides with the smallest strongly convex set with respect to R that contains it. Strongly convex sets are characterized in terms of their associated farthest distance functions, and farthest distance functions associated with strongly convex sets are characterized, too.