Bregman distances and Chebyshev sets

TL;DR

This paper extends the classical characterization of Chebyshev sets in Euclidean spaces to Bregman distances, showing that uniqueness of nearest points implies convexity using nonsmooth analysis and monotone operator theory.

Contribution

It generalizes the convexity characterization of Chebyshev sets from Euclidean to Bregman distances, introducing new analytical approaches.

Findings

Unique nearest point property implies convexity in Bregman distance spaces.

Provides subdifferentiability properties of Bregman nearest distance functions.

Uses nonsmooth analysis and maximal monotone operator theory for proofs.

Abstract

A closed set of a Euclidean space is said to be Chebyshev if every point in the space has one and only one closest point in the set. Although the situation is not settled in infinite-dimensional Hilbert spaces, in 1932 Bunt showed that in Euclidean spaces a closed set is Chebyshev if and only if the set is convex. In this paper, from the more general perspective of Bregman distances, we show that if every point in the space has a unique nearest point in a closed set, then the set is convex. We provide two approaches: one is by nonsmooth analysis; the other by maximal monotone operator theory. Subdifferentiability properties of Bregman nearest distance functions are also given.