On Some Metric in the Family of Compact Convex Sets

• Autorzy: Leśniewski Andrzej , Leśniewska Aleksandra , Szymanek Sylwia

Identyfikatory

DOI

10.7862/rf.2024.3

• Języki publikacji: EN

Abstrakty

EN

We define the Demyanov metric and new metric and compare with the Hausdorff and Vitale metrics. Vitale compared the Hausdorff metric ρH and Vitale metric ρV . We proved that main metric ρLV is equivalence with ρH metric and that the family of nonempty, convex, compact sets and the ρLV metric is the complete space.

Słowa kluczowe

EN

Demyanov metric; Hausdorff metric; convex sets; support function

PL

metryka Demyanova; metryka Hausdorffa; zbiory wypukłe; wsparcie funkcji

• Wydawca: Oficyna Wydawnicza Politechniki Rzeszowskiej
• Czasopismo: Journal of Mathematics and Applications
• Rocznik: 2024
• Tom: Vol. 47
• Strony: 39--45
• Opis fizyczny: Bibliogr. 10 poz.

Twórcy

autor

Leśniewski Andrzej

• Institute of Civil Engineering, University of Live Science, Warsaw, Poland

• https://orcid.org/009-0007-7469-1034

autor

Leśniewska Aleksandra

• Institute of Civil Engineering, University of Live Science, Warsaw, Poland

• https://orcid.org/000-0002-5798-152X

autor

Szymanek Sylwia

• Institute of Civil Engineering, University of Live Science, Warsaw, Poland

• https://orcid.org/000-0001-6181-8541

Bibliografia

• [1] J.P. Aubin, A. Cellina, Differential Inclusions, Grundlehren Math. Wiss. 264, Springer, Berlin, 1984.
• [2] P. Diamond, P. Kloeden, A. Rubinov, A. Vladimirov, Comparative properties of three metrics in the space of compact convex sets, Set-Valued Anal. 5 (3) (1997) 267–289.
• [3] A. Le´sniewski, T. Rze˙zuchowski, The Demyanov metric for convex, bounded sets and existence of Lipschitzian selectors, J. Convex Anal. 18 (3) (2011) 737–747.
• [4] A. Pli´s, Uniqueness of optimal trajectories for non-linear control systems, Ann. Polon. Math. 29 (4) (1975) 397–401.
• [5] T. Rze˙zuchowski, The Demyanov metric and some other metrics in the family of convex sets, Cent. Eur. J. Math. 10 (6) (2012) 2229–2239.
• [6] R. Schneider, Convex Bodies: The Brunn-Minkowski Theory, Encyclopedia Math. Appl. 44, Cambridge University Press, Cambridge, 1993.
• [7] R.A. Vitale, Lp metrics for compact, convex sets, J. Approximation Theory 45 (1985) 280–287.
• [8] J. Sadowski, Young inequality for convolution and its application in convex and set-valued analysis, J. Math. Anal. Appl. (2015) 1274–1294.
• [9] L. Coutin, N. Marie, P.R. de Fitte, On a set-valued Young integral with applications to differential inclusions, hal-03252856v1 (2021).
• [10] A. Le´sniewski, The Demyanov continuous and Cesari’s property, Trans. J. Math. Anal. Appl. 2 (1) (2014).

Uwagi

Opracowanie rekordu ze środków MNiSW, umowa nr POPUL/SP/0154/2024/02 w ramach programu "Społeczna odpowiedzialność nauki II" - moduł: Popularyzacja nauki (2025).