Bregman distances and Klee sets

  • Authors:

  • Heinz H. Bauschke;Xianfu Wang;Jane Ye;Xiaoming Yuan

  • Affiliations:

  • Mathematics, Irving K. Barber School, The University of British Columbia Okanagan, Kelowna, B.C. V1V 1V7, Canada;Mathematics, Irving K. Barber School, The University of British Columbia Okanagan, Kelowna, B.C. V1V 1V7, Canada;Department of Mathematics and Statistics, University of Victoria, Victoria, B.C. V8P 5C2, Canada;Department of Mathematics, Hong Kong Baptist University, PR China

  • Venue:
  • Journal of Approximation Theory
  • Year:
  • 2009

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Abstract

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.