Klee sets and Chebyshev centers for the right Bregman distance

Authors:
Heinz H. Bauschke;Mason S. Macklem;Jason B. Sewell;Xianfu Wang
Affiliations:
Mathematics, Irving K.Barber School, UBC Okanagan, Kelowna, British Columbia V1V 1V7, Canada;Mathematics, Irving K.Barber School, UBC Okanagan, Kelowna, British Columbia V1V 1V7, Canada;Combinatorics & Optimization, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada;Mathematics, Irving K.Barber School, UBC Okanagan, Kelowna, British Columbia V1V 1V7, Canada
Venue:
Journal of Approximation Theory
Year:
2010

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Abstract

We systematically investigate the farthest distance function, farthest points, Klee sets, and Chebyshev centers, with respect to Bregman distances induced by Legendre functions. These objects are of considerable interest in Information Geometry and Machine Learning; when the Legendre function is specialized to the energy, one obtains classical notions from Approximation Theory and Convex Analysis. The contribution of this paper is twofold. First, we provide an affirmative answer to a recently-posed question on whether or not every Klee set with respect to the right Bregman distance is a singleton. Second, we prove uniqueness of the Chebyshev center and we present a characterization that relates to previous works by Garkavi, by Klee, and by Nielsen and Nock.