Abstract

It is shown that an alternative formula for the Demyanov metric permits to extend this metric to the family of convex, bounded sets which need not be closed. Existence of Lipschitzian, linear selectors is discussed.

Author information

Contact details are reproduced from the original publication and may be historical.

Andrzej Leśniewski

Faculty of Mathematics and Information Science, Warsaw University of Technology, Pl. Politechniki 1, 00-661 Warsaw, Poland

Tadeusz Rzeżuchowski

Faculty of Mathematics and Information Science, Warsaw University of Technology, Pl. Politechniki 1, 00-661 Warsaw, Poland

T.Rzezuchowski@mini.pw.edu.pl

Keywords

  • Convex and bounded sets
  • Demyanov metric
  • Hausdorff metric
  • linear selectors

Mathematics Subject Classification

52A20, 54C65, 58C06, 54E35

References

  1. J. P. Aubin, A. Cellina: Differential Inclusions, Springer, Berlin (1984).https://doi.org/10.1007/978-3-642-69512-4Google Scholar
  2. R. Baier, E. Farkhi: Differences of convex compact sets in the space of directed sets. I: The space of directed sets, Set-Valued Analysis 9 (2001) 217–245.https://doi.org/10.1023/A:1012046027626Google Scholar
  3. R. Baier, E. Farkhi: Regularity and integration of set-valued maps represented by generalized Steiner points, Set-Valued Analysis 15 (2007) 185–207.https://doi.org/10.1007/s11228-006-0038-0Google Scholar
  4. S. G. Bartels, D. Pallaschke: Some remarks on the space of differences of sublinear functions, Appl. Math. 22 (1994) 419–426.https://doi.org/10.4064/am-22-3-419-426Google Scholar
  5. V. F. Demyanov, A. M. Rubinov: Constructive Nonsmooth Analysis, Approximation & Optimization 7, Peter Lang, Frankfurt am Main (1995).Google Scholar
  6. V. F. Demyanov, A. M. Rubinov: Quasidifferentiability and Related Topics, Kluwer, Dordrecht (2000).https://doi.org/10.1007/978-1-4757-3137-8Google Scholar
  7. D. Dentcheva: Differentiable selections and Castaing representations of multifunctions, J. Math. Anal. Appl. 223 (1998) 371–396.https://doi.org/10.1006/jmaa.1997.5735Google Scholar
  8. D. Dentcheva: Continuity of multifunctions characterized by Steiner selections, Nonlinear Anal. 47 (2001) 1985–1996.https://doi.org/10.1016/S0362-546X(01)00327-3Google Scholar
  9. P. Diamond, P. Kloeden, A. Rubinov, A. Vladimirov: Comparative properties of three metrics in the space of compact convex sets, Set-Valued Analysis 5 (1997) 267–289.https://doi.org/10.1023/A:1008667909101Google Scholar
  10. A. Pliś: Uniqueness of optimal trajectories for non-linear control systems, Ann. Pol. Math. 29 (1975) 397–401.https://doi.org/10.4064/ap-29-4-397-401Google Scholar
  11. K. Przeslawski: Lipschitz continuous selectors. Part I: Linear selectors, J. Convex Analysis 5 (1998) 249–267.https://doi.org/10.68381/jca05-18Google Scholar
  12. B. Schneider: Convex Bodies: The Brunn-Minkowski Theory, Cambridge University Press, Cambridge (1993).https://doi.org/10.1017/CBO9780511526282Google Scholar

Suggested citation

A. Leśniewski, T. Rzeżuchowski. “The Demyanov Metric for Convex, Bounded Sets and Existence of Lipschitzian Selectors.” Journal of Convex Analysis 18 (2011), No. 3, 737–747. https://doi.org/10.68381/jca18047

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Published by Heldermann Verlag, 2011. Rights now held by Banach Press.