Abstract of a paper by Heinz H. Bauschke, Xianfu Wang, Jane Ye and Xiaoming Yuan
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.
participants (1)
-
alspach@fourier.math.okstate.edu