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The Demyanov metric and some other metrics in the family of convex sets. (English) Zbl 1364.52004

Summary: We describe some known metrics in the family of convex sets which are stronger than the Hausdorff metric and propose a new one. These stronger metrics preserve in some sense the facial structure of convex sets under small changes of sets.

MSC:

52A20 Convex sets in \(n\) dimensions (including convex hypersurfaces)
54E50 Complete metric spaces
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[1] Aubin J.-P., Cellina A., Differential Inclusions, Grundlehren Math. Wiss., 264, Springer, Berlin, 1984 http://dx.doi.org/10.1007/978-3-642-69512-4
[2] Baier R., Farkhi E.M., Differences of convex compact sets in the space of directed sets I. The space of directed sets, Set-Valued Anal., 2001, 9(3), 217-245 http://dx.doi.org/10.1023/A:1012046027626 · Zbl 1097.49507
[3] Demyanov V.F., Rubinov A.M., Constructive Nonsmooth Analysis, Approximation & Optimization, 7, Peter Lang, Frankfurt am Main, 1995 · Zbl 0887.49014
[4] Demyanov V.F., Rubinov A.M. (Eds.), Quasidifferentiability and Related Topics, Nonconvex Optim. Appl., 43, Kluwer, Dordrecht, 2000
[5] Diamond P., Kloeden P., Rubinov A., Vladimirov A., Comparative properties of three metrics in the space of compact convex sets, Set-Valued Anal., 1997, 5(3), 267-289 http://dx.doi.org/10.1023/A:1008667909101
[6] Grzybowski J., Lesniewski A., Rzezuchowski T., The completion of the space of convex, bounded sets with respect to the Demyanov metric, Demonstratio Math. (in press) · Zbl 1290.52004
[7] Lesniewski A., Rzezuchowski T., The Demyanov metric for convex, bounded sets and existence of Lipschitzian Selectors, J. Convex Anal., 2011, 18(3), 737-747 · Zbl 1227.52002
[8] Plis A., Uniqueness of optimal trajectories for non-linear control systems, Ann. Polon. Math., 1975, 29(4), 397-401 · Zbl 0316.49029
[9] Schneider R., Convex Bodies: The Brunn-Minkowski Theory, Encyclopedia Math. Appl., 44, Cambridge University Press, Cambridge, 1993 http://dx.doi.org/10.1017/CBO9780511526282 · Zbl 0798.52001
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