Chebyshev centers that are not farthest points

TL;DR

This paper investigates the relationship between Chebyshev centers and farthest points in Banach spaces, revealing how convexity properties influence whether a Chebyshev center can be a farthest point.

Contribution

It provides a characterization of strictly convex Banach spaces based on the behavior of Chebyshev centers and farthest points, including new insights into uniformly convex spaces.

Findings

In two-dimensional real strictly convex spaces, Chebyshev centers are not farthest points.

In uniformly convex Banach spaces, the relationship between centers and farthest points is characterized.

Examples demonstrate the optimality of the theoretical results.

Abstract

In this paper we address the question whether in a given Banach space, a Chebyshev center of a nonempty bounded subset can be a farthest point of the set. Our exploration reveals that the answer depends on the convexity properties of the Banach space. We obtain a characterization of two-dimensional real strictly convex spaces in terms of Chebyshev center not contributing to the set of farthest points. We explore the scenario in uniformly convex Banach spaces and further study the roles played by centerability and M-compactness in the scheme of things to obtain a step by step characterization of strictly convex Banach spaces. We also illustrate with examples the optimality of our results.