This is an announcement for the paper "Bregman distances and Klee sets" by Heinz H. Bauschke, Xianfu Wang, Jane Ye and Xiaoming Yuan.
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.
Archive classification: math.FA math.OC
Mathematics Subject Classification: 47H05; 41A65; 49J52
The source file(s), submitted.tex: 49600 bytes, is(are) stored in gzipped form as 0802.2322.gz with size 15kb. The corresponding postcript file has gzipped size 113kb.
Submitted from: heinz.bauschke@ubc.ca
The paper may be downloaded from the archive by web browser from URL
http://front.math.ucdavis.edu/0802.2322
or
http://arXiv.org/abs/0802.2322
or by email in unzipped form by transmitting an empty message with subject line
uget 0802.2322
or in gzipped form by using subject line
get 0802.2322
to: math@arXiv.org.