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 previousworks by Garkavi, by Klee, and byNielsen andNock. 2000 Mathematics Subject Classification: Primary 41A65, 46B20; Secondary 47H05, 49J52, 49J53, 90C25.