Bregman distances and Chebyshev sets

Heinz H. Bauschke*, Xianfu Wang, Jane Ye, Xiaoming YUAN

*Corresponding author for this work

Research output: Contribution to journalJournal articlepeer-review

32 Citations (Scopus)


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.

Original languageEnglish
Pages (from-to)3-25
Number of pages23
JournalJournal of Approximation Theory
Issue number1
Publication statusPublished - Jul 2009

Scopus Subject Areas

  • Analysis
  • Numerical Analysis
  • Mathematics(all)
  • Applied Mathematics

User-Defined Keywords

  • Bregman distance
  • Bregman projection
  • Chebyshev set with respect to a Bregman distance
  • Legendre function
  • Maximal monotone operator
  • Nearest point
  • Subdifferential operators


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