Bregman distances and Klee sets

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

*Corresponding author for this work

Research output: Contribution to journalJournal articlepeer-review

9 Citations (Scopus)


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 the work by Hiriart-Urruty on the Euclidean case.

Original languageEnglish
Pages (from-to)170-183
Number of pages14
JournalJournal of Approximation Theory
Issue number2
Publication statusPublished - Jun 2009

Scopus Subject Areas

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

User-Defined Keywords

  • Bregman distance
  • Bregman projection
  • Convex function
  • Farthest point
  • Legendre function
  • Maximal monotone operator
  • Subdifferential operator


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