Chebyshev Sets, Klee Sets, and Chebyshev Centers with respect to Bregman Distances: Recent Results and Open Problems

TL;DR

This paper reviews recent advances and open problems in the study of Chebyshev and Klee sets, as well as Chebyshev centers, when distances are measured by Bregman divergences rather than traditional norms.

Contribution

It presents new results on Chebyshev and Klee sets with respect to Bregman distances and discusses their connections to Chebyshev functions and open research questions.

Findings

New results on Klee sets and Chebyshev centers with Bregman distances

Connections established between Bregman divergence-based sets and Chebyshev functions

Identification of open problems in the geometric analysis of Bregman distances

Abstract

In Euclidean spaces, the geometric notions of nearest-points map, farthest-points map, Chebyshev set, Klee set, and Chebyshev center are well known and well understood. Since early works going back to the 1930s, tremendous theoretical progress has been made, mostly by extending classical results from Euclidean space to Banach space settings. In all these results, the distance between points is induced by some underlying norm. Recently, these notions have been revisited from a different viewpoint in which the discrepancy between points is measured by Bregman distances induced by Legendre functions. The associated framework covers the well known Kullback-Leibler divergence and the Itakura-Saito distance. In this survey, we review known results and we present new results on Klee sets and Chebyshev centers with respect to Bregman distances. Examples are provided and connections to recent work on Chebyshev functions are made. We also identify several intriguing open problems.