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.


52A20 Convex sets in \(n\) dimensions (including convex hypersurfaces)
54E50 Complete metric spaces
Full Text: DOI


[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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.