Bregman distances and Chebyshev sets

Authors:
Heinz H. Bauschke;Xianfu Wang;Jane Ye;Xiaoming Yuan
Affiliations:
Mathematics, Irving K. Barber School, The University of British Columbia Okanagan, Kelowna, B.C. V1V 1V7, Canada;Mathematics, Irving K. Barber School, The University of British Columbia Okanagan, Kelowna, B.C. V1V 1V7, Canada;Department of Mathematics and Statistics, University of Victoria, Victoria, B.C. V8W 3P4, Canada;Department of Mathematics, Hong Kong Baptist University, PR China
Venue:
Journal of Approximation Theory
Year:
2009

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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.