The completion of the space of convex, bounded sets with respect to the Demyanov metric

• Autorzy: Grzybowski, J. , Leśniewski, A. , Rzeżuchowski, T.
• Języki publikacji: EN

Abstrakty

EN

The Demyanov metric in the family of convex, compact sets in finite dimensional spaces has been recently extended to the family of convex, bounded sets – not necessarily closed. In this note it is shown that these spaces are not complete and a model for the completion is proposed. A full answer is given in R2 and the situation in higher dimensions is discussed.

Słowa kluczowe

PL

metryka Demyanova; metryka Hausdorffa; przestrzeń metryczna

EN

convex sets; bounded sets; Demyanov metric; Hausdorff metric; metric space

• Czasopismo: Demonstratio Mathematica
• Rocznik: 2013
• Tom: Vol. 46, nr 1
• Strony: 191--196
• Opis fizyczny: Bibliogr. 7 poz.

Twórcy

autor

Grzybowski, J.

• Faculty of Mathematics & Computer Science, Adam Mickiewicz University, ul. Umultowska 87, 61-164 Poznań, Poland

autor

Leśniewski, A.

• Faculty of Mathematics & Information Science, Warsaw University of Technology, Koszykowa 75, 00-662 Warsaw, Poland

autor

Rzeżuchowski, T.

• Faculty of Mathematics & Information Science, Warsaw University of Technology, Koszykowa 75, 00-662 Warsaw, Poland,

Bibliografia

• [1] J. P. Aubin, A. Cellina, Differential Inclusions, Springer-Verlag, Germany, 1984.
• [2] R. Baier, E. Farkhi, Differences of convex compact sets in the space of directed sets. Part I: The space of directed sets, Set-Valued Anal. 9 (2001), 217–245.
• [3] V. F. Demyanov, A. M. Rubinov, Quasidifferentiability and Related Topics, Kluwer, Netherlands, 2000.
• [4] V. F. Demyanov, A. M. Rubinov, Constructive Nonsmooth Analysis, vol. 7 of Approximation & Optimization, Peter Lang, Frankfurt, 1995.
• [5] P. Diamond, P. Kloeden, A. Rubinov, A. Vladimirov, Comparative properties of three metrics in the space of compact convex sets, Set-Valued Anal. 5 (1997), 267–289.
• [6] A. Lesniewski, T. Rzezuchowski, The Demyanov metric for convex, bounded sets and existence of Lipschitzian selectors, to appear in J. Conv. Anal. 18 (2011), No. 3.
• [7] R. Schneider, Convex Bodies: The Brunn-Minkowski Theory, Cambridge University Press, 1993.