This is an announcement for the paper "Bregman distances and Chebyshev sets" by Heinz H. Bauschke, Xianfu Wang, Jane Ye, and Xiaoming Yuan.

Abstract: A closed set of a Euclidean space is said to be Chebyshev if every point in the space has one and only one closest point in the set. Although the situation is not settled in infinite-dimensional Hilbert spaces, in 1932 Bunt showed that in Euclidean spaces a closed set is Chebyshev if and only if the set is convex. In this paper, from the more general perspective of Bregman distances, we show that if every point in the space has a unique nearest point in a closed set, then the set is convex. We provide two approaches: one is by nonsmooth analysis; the other by maximal monotone operator theory. Subdifferentiability properties of Bregman nearest distance functions are also given.

Archive classification: math.FA

Mathematics Subject Classification: Primary 41A65; Secondary 47H05, 49J52.

The source file(s), submitted.tex: 67922 bytes, is(are) stored in gzipped form as 0712.4030.gz with size 19kb. The corresponding postcript file has gzipped size 134kb.

Submitted from: heinz.bauschke@ubc.ca

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http://arXiv.org/abs/0712.4030

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