Chebyshev Sets, Klee Sets, and Chebyshev Centers with respect to Bregman Distances: Recent Results and Open Problems

摘要

In Euclidean spaces, the geometric notions of nearest-points map,farthest-points map, Chebyshev set, Klee set, and Chebyshev center are wellknown and well understood. Since early works going back to the 1930s,tremendous theoretical progress has been made, mostly by extending classicalresults from Euclidean space to Banach space settings. In all these results,the distance between points is induced by some underlying norm. Recently, thesenotions have been revisited from a different viewpoint in which the discrepancybetween points is measured by Bregman distances induced by Legendre functions.The associated framework covers the well known Kullback-Leibler divergence andthe Itakura-Saito distance. In this survey, we review known results and wepresent new results on Klee sets and Chebyshev centers with respect to Bregmandistances. Examples are provided and connections to recent work on Chebyshevfunctions are made. We also identify several intriguing open problems.