Bregman distances and Klee sets

TL;DR

This paper investigates Klee sets using Bregman distances instead of Euclidean norms, proving that under certain conditions, such sets must be singletons, extending classical results to a broader context.

Contribution

It introduces a new perspective on Klee sets with Bregman distances and proves they are singletons under specific conditions, using Monotone Operator Theory and Nonsmooth Analysis.

Findings

Klee sets with Bregman distances are singletons under certain conditions.

Two different proofs are provided, one based on Monotone Operator Theory and another on Nonsmooth Analysis.

Results extend classical Euclidean Klee set results to Bregman distance settings.

Abstract

In 1960, Klee showed that a subset of a Euclidean space must be a singleton provided that each point in the space has a unique farthest point in the set. This classical result has received much attention; in fact, the Hilbert space version is a famous open problem. In this paper, we consider Klee sets from a new perspective. Rather than measuring distance induced by a norm, we focus on the case when distance is meant in the sense of Bregman, i.e., induced by a convex function. When the convex function has sufficiently nice properties, then - analogously to the Euclidean distance case - every Klee set must be a singleton. We provide two proofs of this result, based on Monotone Operator Theory and on Nonsmooth Analysis. The latter approach leads to results that complement work by Hiriart-Urruty on the Euclidean case.