2010/03/16 by Heinz H. Bauschke, Bauschke, Heinz H., Mason S. Macklem +3 · 1 citation
Decision Sciences · Mathematics · #Advanced Statistical Methods and Models #FOS: Mathematics #Mathematical Inequalities and Applications #Multi-Criteria Decision Making #Optimization and Control (math.OC) #math.OC
paper · pdf · doi:10.48550/arxiv.1003.3127
arxiv created 2010/03/16 · openalex publication_date 2010/03/16 · arxiv updated 2010/03/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In Euclidean spaces, the geometric notions of nearest-points map, farthest-points map, Chebyshev set, Klee set, and Chebyshev center are well known and well understood. Since early works going back to the 1930s, tremendous theoretical progress has been made, mostly by extending classical results from Euclidean space to Banach space settings. In all these results, the distance between points is induced by some underlying norm. Recently, these notions have been revisited from a different viewpoint in which the discrepancy between points is measured by Bregman distances induced by Legendre functions. The associated framework covers the well known Kullback-Leibler divergence and the Itakura-Saito distance. In this survey, we review known results and we present new results on Klee sets and Chebyshev centers with respect to Bregman distances. Examples are provided and connections to recent work on Chebyshev functions are made. We also identify several intriguing open problems.