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arXiv · 0802.2322

Bregman distances and Klee sets

Heinz H. Bauschke·Xianfu Wang·Jane Ye·Xiaoming Yuan
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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 work by Hiriart-Urruty on the Euclidean case.

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Heinz H. Bauschke, Xianfu Wang, Jane Ye, Xiaoming Yuan. 2008-02-16. Bregman distances and Klee sets. https://arxiv.org/abs/0802.2322

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