Abstract
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.
AMS 2010 Subject Classification: Primary 41A65; Secondary 28D05, 41A50, 46N10, 47N10, 49J53, 54E52, 58C06, 90C25
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Asplund, E.: Sets with unique farthest points. Israel J. Math. 5, 201–209 (1967)
Bauschke, H.H., Borwein, J.M.: Legendre functions and the method of random Bregman projections. J. Convex Anal. 4, 27–67 (1997)
Bauschke, H.H., Borwein, J.M.: Joint and separate convexity of the Bregman distance. In: D. Butnariu, Y. Censor, S. Reich (ed.) Inherently Parallel Algorithms in Feasibility and Optimization and their Applications (Haifa 2000), pp. 23–36. Elsevier (2001)
Bauschke, H.H., Noll, D.: The method of forward projections. J. Nonlin. Convex Anal. 3, 191–205 (2002)
Bauschke, H.H., Borwein, J.M, Combettes, P.L.: Essential smoothness, essential strict convexity, and Legendre functions in Banach spaces. Commun. Contemp. Math. 3, 615–647 (2001)
Bauschke, H.H., Wang, X., Ye, J., Yuan, X.: Bregman distances and Chebyshev sets. J. Approx. Theory 159, 3–25 (2009)
Bauschke, H.H., Wang, X., Ye, J., Yuan, X.: Bregman distances and Klee sets. J. Approx. Theory 158, 170–183 (2009)
Bauschke, H.H., Macklem, M.S., Sewell, J.B., Wang, X.: Klee sets and Chebyshev centers for the right Bregman distance. J. Approx. Theory 162, 1225–1244 (2010)
Berens, H., Westphal, U.: Kodissipative metrische Projektionen in normierten linearen Räumen. In: P. L. Butzer and B. Sz.-Nagy (eds.) Linear Spaces and Approximation, vol. 40, pp. 119–130, Birkhäuser (1980)
Borwein, J.M.: Proximity and Chebyshev sets. Optim. Lett. 1, 21–32 (2007)
Borwein, J.M, Lewis, A.S.: Convex Analysis and Nonlinear Optimization, 2nd edn. Springer (2006)
Borwein, J.M., Vanderwerff, J.: Convex Functions: Constructions, Characterizations and Counterexamples. Cambridge University Press (2010)
Bregman, L.M.: The relaxation method of finding the common point of convex sets and its application to the solution of problems in convex programming. U.S.S.R. Comp. Math. Math 7, 200–217 (1967)
Bunt, L.N.H.: Bijdrage tot de theorie de convexe puntverzamelingen. Thesis, Univ. of Groningen, Amsterdam, 1934
Butnariu, D., Iusem, A.N.: Totally Convex Functions for Fixed Points Computation in Infinite Dimensional Optimization. Kluwer, Dordrecht (2000)
Censor, Y., Zenios, S.A.: Parallel Optimization. Oxford University Press (1997)
Deutsch, F.: Best Approximation in Inner Product Spaces. Springer (2001)
Fremlin, D.H.: Measure Theory, vol. 2. Broad Foundations, 2nd edn. Torres Fremlin, Colchester (2010)
Garkavi, A.L.: On the Čebyšev center and convex hull of a set. Usp. Mat. Nauk 19, 139–145 (1964)
Goebel, K., Kirk, W.A.: Topics in Metric Fixed Point Theory. Cambridge University Press (1990)
De Guzmán, M.: A change-of-variables formula without continuity. Am. Math. Mon. 87, 736–739 (1980)
Hiriart-Urruty, J.-B.: Ensembles de Tchebychev vs. ensembles convexes: l’etat de la situation vu via l’analyse convexe non lisse. Ann. Sci. Math. Québec 22, 47–62 (1998)
Hiriart-Urruty, J.-B.: La conjecture des points les plus éloignés revisitée. Ann. Sci. Math. Québec 29, 197–214 (2005)
Hiriart-Urruty, J.-B.: Potpourri of conjectures and open questions in nonlinear analysis and optimization. SIAM Rev. 49, 255–273 (2007)
Hiriart-Urruty, J.-B., Lemaréchal, C.: Convex Analysis and Minimization Algorithms II. Springer (1996)
Klee, V.: Circumspheres and inner products. Math. Scand. 8, 363–370 (1960)
Klee, V.: Convexity of Chebyshev sets. Math. Ann. 142, 292–304 (1960/1961)
Motzkin, T.: Sur quelques propriétés caractéristiques des ensembles convexes. Atti. Accad. Naz. Lincei, Rend., VI. Ser. 21, 562–567 (1935)
Motzkin, T.S., Straus, E.G., Valentine, F.A.: The number of farthest points. Pac. J. Math. 3, 221–232 (1953)
Nielsen, F., Nock, R.: On the smallest enclosing information disk. Inform. Process. Lett. 105, 93–97 (2008)
Nock, R., Nielsen, F.: Fitting the smallest enclosing Bregman ball. In: J. Gama, R. Camacho, P. Brazdil, A. Jorge and L. Torgo (eds.) Machine Learning: 16th European Conference on Machine Learning (Porto 2005), pp. 649–656, Springer Lecture Notes in Computer Science vol. 3720 (2005)
Reich, S., Sabach, S.: Two strong convergence theorems for Bregman strongly nonexpansive operators in reflexive Banach spaces. Nonlinear Anal. 73, 122–135 (2010)
Rockafellar, R.T.: Convex Analysis. Princeton University Press, Princeton (1970)
Rockafellar, R.T., Wets, R. J.-B.: Variational Analysis. Springer, New York (1998)
Singer, I.: Best Approximation in Normed Linear Spaces by Elements of Linear Subspaces. Springer (1970)
Singer, I.: The Theory of Best Approximation and Functional Analysis. Conference Board of the Mathematical Sciences Regional Conference Series in Applied Mathematics, No. 13. Society for Industrial and Applied Mathematics (1974)
Vlasov, L.P.: Approximate properties of sets in normed linear spaces. Russian Math. Surv. 28, 1–66 (1973)
Wang, X.: On Chebyshev functions and Klee functions. J. Math. Anal. Appl. 368, 293–310 (2010)
Westphal, U., Schwartz, T.: Farthest points and monotone operators. B. Aust. Math. Soc. 58, 75–92 (1998)
Zălinescu, C.: Convex Analysis in General Vector Spaces. World Scientific Publishing (2002)
Acknowledgements
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.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2011 Springer Science+Business Media, LLC
About this chapter
Cite this chapter
Bauschke, H.H., Macklem, M.S., Wang, X. (2011). Chebyshev Sets, Klee Sets, and Chebyshev Centers with Respect to Bregman Distances: Recent Results and Open Problems. In: Bauschke, H., Burachik, R., Combettes, P., Elser, V., Luke, D., Wolkowicz, H. (eds) Fixed-Point Algorithms for Inverse Problems in Science and Engineering. Springer Optimization and Its Applications(), vol 49. Springer, New York, NY. https://doi.org/10.1007/978-1-4419-9569-8_1
Download citation
DOI: https://doi.org/10.1007/978-1-4419-9569-8_1
Published:
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4419-9568-1
Online ISBN: 978-1-4419-9569-8
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)