Bregman距离和Klee集
Bregman distances and Klee sets
摘要：
在1960年，克利表明，一个子集的欧几里得空间必须是一个单点提供的每一个点在空间有一个独特的最远点的设置。这一经典结果受到了广泛关注;事实上，希尔伯特空间版本是一个著名的开放问题。本文从一个新的角度考虑Klee集。而不是测量由规范引起的距离，我们专注于距离在Bregman意义上的情况，即，由凸函数诱导。当凸函数具有足够好的性质时，那么--类似于欧几里得距离的情况--每个克利集都必须是单点集。我们给出了这个结果的两个证明，基于单调算子理论和非光滑分析。后一种方法导致的结果，补充工作Hiriart-Urruty的欧几里德的情况下。
英文摘要：
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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