Bregman distances and Chebyshev sets

2007/12/24 by Heinz H. Bauschke, Xianfu Wang, Bauschke, Heinz H. +6 · 3 citations
Computer Science · Mathematics · #41A65 (Primary) #47H05 #49J52 (Secondary) #FOS: Mathematics #Functional Analysis (math.FA) #Functional Equations Stability Results #Fuzzy Systems and Optimization #Optimization and Variational Analysis #math.FA #msc:41A65 #msc:47H05 #msc:49J52

paper · pdf · doi:10.48550/arxiv.0712.4030

arxiv created 2007/12/24 · openalex publication_date 2007/12/24 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

Cited by

• Alternating Bregman projections and convergence of the EM algorithm
• Bregman level proximal subdifferentials and new characterizations of Bregman proximal operators
• On the natural domain of Bregman operators

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