On the Convergence of Sequences of Convex Sets in Finite Dimensions

摘要

Previous article Next article On the Convergence of Sequences of Convex Sets in Finite DimensionsGabriella Salinetti and Roger J.-B. WetsGabriella Salinetti and Roger J.-B. Wetshttps://doi.org/10.1137/1021002PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAboutAbstractFour types of convergence for sequences of convex sets are investigated. Their interrelationships are explored.[1] Casimir Kuratowski, Topologie. Vol. I, Monografie Matematyczne, Tom 20, Państwowe Wydawnictwo Naukowe, Warsaw, 1958xiii+494, Poland MR0090795 0078.14603 Google Scholar[2] Umberto Mosco, Convergence of convex sets and of solutions of variational inequalities, Advances in Math., 3 (1969), 510–585 10.1016/0001-8708(69)90009-7 MR0298508 0192.49101 CrossrefISIGoogle Scholar[3] B. Van Cutsem, Masters Thesis, Eléments aléatoires à valeurs convexes compactes, Thèse, Grenoble, France, 1971 Google Scholar[4] George B. Dantzig, , Jon Folkman and , Norman Shapiro, On the continuity of the minimum sets of a continuous function, J. Math. Anal. Appl., 17 (1967), 519–548 10.1016/0022-247X(67)90139-4 MR0207426 0153.49201 CrossrefISIGoogle Scholar[5] F. Hausdorff, Grundzüge der Mengenlehre, Verlag von Veit, Leipzig, 1914, Reprinted by Chelsea, New York 45.0123.01 Google Scholar[6] Tosio Kato, Perturbation theory for linear operators, Die Grundlehren der mathematischen Wissenschaften, Band 132, Springer-Verlag New York, Inc., New York, 1966xix+592 MR0203473 0148.12601 CrossrefGoogle Scholar[7] David W. Walkup and , Roger J.-B. Wets, Continuity of some convex-cone-valued mappings, Proc. Amer. Math. Soc., 18 (1967), 229–235 MR0209806 0145.38004 CrossrefGoogle Scholar[8] Bernard Van Cutsem, Martingales de convexes fermés aléatoires en dimension finie, Ann. Inst. H. Poincaré Sect. B (N.S.), 8 (1972), 365–385 MR0378088 0252.60022 Google Scholar[9] Gabriella Salinetti and , Roger J.-B. Wets, On the relations between two types of convergence for convex functions, J. Math. Anal. Appl., 60 (1977), 211–226 10.1016/0022-247X(77)90060-9 MR0479398 0359.54005 CrossrefISIGoogle Scholar[10] R. A. Wijsman, Convergence of sequences of convex sets, cones and functions. II, Trans. Amer. Math. Soc., 123 (1966), 32–45 MR0196599 0146.18204 CrossrefISIGoogle Scholar[11] R. Tyrrell Rockafellar, Convex analysis, Princeton Mathematical Series, No. 28, Princeton University Press, Princeton, N.J., 1970xviii+451 MR0274683 0193.18401 CrossrefGoogle Scholar[12] Lars Hörmander, Sur la fonction d'appui des ensembles convexes dans un espace localement convexe, Ark. Mat., 3 (1955), 181–186 MR0068112 0064.10504 CrossrefGoogle Scholar[13] Jean-Jacques Moreau, Multiapplications à retraction finie, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 1 (1974), 169–203 (1975) MR0377812 0306.54024 Google Scholar[14] Umberto Mosco, On the continuity of the Young-Fenchel transform, J. Math. Anal. Appl., 35 (1971), 518–535 10.1016/0022-247X(71)90200-9 MR0283586 0253.46086 CrossrefISIGoogle Scholar[15] J. L. Joly, Masters Thesis, Une famille de topologies et de convergence sur l'ensemble des fonctionelles convexes, Thèse, Grenoble, France, 1970 Google Scholar[16] Raoul Robert, Convergence de fonctionnelles convexes, J. Math. Anal. Appl., 45 (1974), 533–555 10.1016/0022-247X(74)90050-X MR0352927 0299.46014 CrossrefISIGoogle Scholar[17] E. Kalai, Cooperative non-sidepayment games: Extensions of sidepayment game solutions, metrics and representative functions, Tech. Rep., Center for Applied Math., Cornell Univ., Ithaca, NY, 1972 Google Scholar Previous article Next article FiguresRelatedReferencesCited ByDetails Limits of Eventual Families of Sets with Application to Algorithms for the Common Fixed Point ProblemSet-Valued and Variational Analysis, Vol. 30, No. 3 | 23 March 2022 Cross Ref Properties of Statistical Depth with Respect to Compact Convex Random Sets: The Tukey DepthMathematics, Vol. 10, No. 15 | 3 August 2022 Cross Ref Extended gradient of convex function and capital allocationEuro