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

  • Heinz H. Bauschke
  • Mason S. Macklem
  • Xianfu Wang
Part of the Springer Optimization and Its Applications book series (SOIA, volume 49)


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.


Bregman distance Chebyshev center Chebyshev function Chebyshev point of a function Chebyshev set Convex function Farthest point Fenchel conjugate Itakura–Saito distance Klee set Klee function Kullback–Leibler divergence Legendre function Nearest point Projection 


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The authors thank two referees for their careful reading and pertinent comments. Heinz Bauschke was partially supported by the Natural Sciences and Engineering Research Council of Canada and by the Canada Research Chair Program. Xianfu Wang was partially supported by the Natural Sciences and Engineering Research Council of Canada.


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Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  • Heinz H. Bauschke
    • 1
  • Mason S. Macklem
  • Xianfu Wang
  1. 1.Department of Mathematics, Irving K. Barber SchoolUniversity of British ColumbiaKelownaCanada

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