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Bregman distances and Chebyshev sets
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
Klee sets and Chebyshev centers for the right Bregman distance
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.Comment: 23 pages, 2 figures, 14 image
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