Abstract
A net 〈A λ〉 of nonempty closed sets in a metric space 〈X, d〉 is declaredWijsman convergent to a nonempty closed setA provided for eachx εX, we haved(x, A)=limλ d(x, A). Interest in this convergence notion originates from the seminal work of R. Wijsman, who showed in finite dimensions that the conjugate map for proper lower semicontinuous convex functions preserves convergence in this sense, where functions are identified with their epigraphs. In this paper, we review the attempts over the last 25 years to produce infinite-dimensional extensions of Wijsman's theorem, and we look closely at the topology of Wijsman convergence in an arbitrary metric space as well. Special emphasis is given to the developments of the past five years, and several new limiting counterexamples are presented.
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References
Attouch, H.:Variational Convergence for Functions and Operators, Pitman, New York, 1984.
Attouch, H., Azé, D., and Beer, G.: On some inverse problems for the epigraphical sum,Nonlinear Anal. 16 (1991), 241–254.
Attouch, H., Lucchetti, R., and Wets, R.: The topology of the ρ-Hausdorff distance,Annal. Mat. Pura. Appl. 160 (1991), 303–320.
Attouch, H. and Wets, R.: Isometries for the Legendre-Fenchel transform,Trans. Amer. Math. Soc. 296 (1986), 33–60.
Attouch, H. and Wets, R.: Quantitative stability of variational systems: I. The epigraphical distance,Trans. Amer. Math. Soc. 328 (1991), 695–730.
Azé, D.: Caractérisation de la convergence au sens de Mosco en terme d'approximation infconvolutives,Ann. Fac. Sci. Toulouse 8, (1986–1987), 293–314.
Azé, D. and Penot, J.-P.: Operations on convergent families of sets and functions,Optimization 21 (1990), 521–534.
Azé, D. and Penot, J.-P.: Qualitative results about the convergence of convex sets and convex functions, inOptimization and Nonlinear Analysis (Haifa, 1990), Res. Notes Math. Ser. 244, Longman, Harlow, 1992, pp. 1–24.
J.-P. Aubin and Frankowska, H.:Set-Valued Analysis, Birkhäuser, Boston, 1990.
Baronti, M. and Papini, P.-L.: Convergence of sequences of sets, inMethods of Functional Analysis in Approximation Theory, ISNM 76, Birkhäuser-Verlag, Basel, 1986.
Beer, G.: Metric spaces with nice closed balls and distance functions for closed sets,Bull. Austral. Math. Soc. 35 (1987), 81–96.
Beer, G.: An embedding theorem for the Fell topology,Michigan Math. J. 35 (1988), 3–9.
Beer, G.: Support and distance functionals for convex sets,Numer. Func. Anal. Optim. 10 (1989), 15–36.
Beer, G.: Convergence of continuous linear functionals and their level sets,Archiv. Math. 52 (1989), 482–491.
Beer, G.: Conjugate convex functions and the epi-distance topology,Proc. Amer. Math. Soc. 108 (1990), 117–126.
Beer, G.: Mosco convergence and weak topologies for convex sets and functions,Mathematika 38 (1991), 89–104.
Beer, G.: A Polish topology for the closed subsets of a Polish space,Proc. Amer. Math. Soc. 113 (1991), 1123–1133.
Beer, G.: Topologies on closed and closed convex sets and the Effros measurability of set valued functions,Sém. d'Anal. Convexe Montpellier (1991), exposé No 2.
Beer, G.: The slice topology: A viable alternative to Mosco convergence in nonreflexive spaces,Sém. d'Anal. Convexe Montpellier (1991), exposé No 3;Nonlinear Anal. 19 (1992), 271–290.
Beer, G.: Wijsman convergence of convex sets under renorming,Nonlinear Anal. 22 (1994), 207–216.
Beer, G.: Lipschitz regularization and the convergence of convex functions,Numer. Funct. Anal. Optim. 15 (1994), 31–46.
Beer, G. and Borwein, J.: Mosco convergence and reflexicity,Proc. Amer. Math. Soc. 109 (1990), 427–436.
Beer, G.: Mosco and slice convergence of level sets and graphs of linear functionals,J. Math. Anal. Appl. 175 (1993), 53–67.
Beer, G. and DiConcilio, A.: Uniform convergence on bounded sets and the Attouch-Wets topology,Proc. Amer. Math. Soc. 112 (1991), 235–243.
Beer, G., Lechicki, A., Levi, S., and Naimpally, S.: Distance functionals and the suprema of hyperspace topologies,Annal. Mat. Pura Appl. 162 (1992), 367–381.
Beer, G. and Lucchetti, R.: Weak topologies for the closed subsets of a metrizable space,Trans. Amer. Math. Soc. 335 (1993), 805–822.
Beer, G. and Lucchetti, R.: Well-posed optimization problems and a new topology for the closed subsets of a metric space,Rocky Mountain J. Math. 23 (1993), 1197–1220.
Beer, G. and Pai, D.: On convergence of convex sets and relative Chebyshev centers,J. Approx. Theory 62 (1990), 147–169.
Borwein, J. and Fabian, M.: On convex functions having points of Gateaux differentiability which are not points of Frechet differentiability, Preprint.
Borwein, J. and Fitzpatrick, S.: Mosco convergence and the Kadec property,Proc. Amer. Math. Soc. 106 (1989), 843–852.
Borwein, J. and Vanderwerff, J.: Dual Kadec-Klee norms and the relationship between Wijsman, slice and Mosco convergence, Preprint, University of Waterloo.
Castaing, C. and Valadier, M.:Convex Analysis and Measurable Multifunctions, Lecture Notes in Math. 580, Springer-Verlag, Berlin, 1977.
Corradini, P.: Topologie sull' iperspazio di uno spazio lineare normato, Tesi di laurea, Universitá degli studi di Milano, 1991.
Costantini, C., Levi, S., and Zieminska, J.: Metrics that generate the same hyperspace convergence,Set-Valued Analysis 1 (1993), 141–157.
Cornet, B.: Topologies sur les fermés d'un espace métrique, Cahiers de mathématiques de la décision 7309, Université de Paris Dauphine, 1973.
Del Prete, I. and Lignola, B.: On the convergence of closed-valued multifunctions,Boll. Un. Mat. Ital. B 6 (1983), 819–834.
Diestel, J.:Geometry of Banach Spaces — Selected Topics, Lecture Notes in Math. 485, Springer-Verlag, Berlin, 1975.
Di Maio, G. and Naimpally, S.: Comparison of hypertopologies,Rend. Istit. Mat. Univ. Trieste 22 (1990), 140–161.
Effros, E.: Convergence of closed subsets in a topological space,Proc. Amer. Math. Soc. 16 (1965), 929–931.
Engelking, R.:General Topology, Polish Scientific Publishers, Warsaw, 1977.
Fell, J.: A Hausdorff topology for the closed subsets of a locally compact non-Hausdorff space,Proc. Amer. Math. Soc. 13 (1962), 472–476.
Francaviglia, S., Lechicki, A., and Levi, S.: Quasi-uniformizationn of hyperspaces and convergence of nets of semicontinuous multifunctions,J. Math. Anal. Appl. 112 (1985), 347–370.
Hess, C.: Contributions à l'étude de la mesurabilité, de la loi de probabilité, et de la convergence des multifunctions, Thèse d'état, Montpellier, 1986.
Hiriart-Urruty, J.-B.: Lipschitzr-continuity of the approximate subdifferential of a convex function,Math. Scand. 47 (1980), 123–134.
Holá, L. and Lucchetti, R.: Comparison of hypertopologies, Preprint.
Holmes, R.:Geometric Functional Analysis, Springer-Verlag, New York, 1975.
Hörmander, L.: Sur la fonction d'appui des ensembles convexes dans une espace localement convexe,Arkiv. Mat. 3 (1954), 181–186.
Joly, J.: Une famille de topologies sur l'ensemble des fonctions convexes pour lesquelles la polarité est bicontinue,J. Math. Pures Appl. 52 (1973), 421–441.
Klein, E. and Thompson, A.:Theory of Correspondences, Wiley, New York, 1984.
Kuratowski, K.:Topology, vol. 1, Academic Press, New York, 1966.
M. Lavie: Contribution a l'étude de la convergence de sommes d'ensembles aléatoires indépendants et martingales multivoques, Thèse, Montpellier, 1990.
Lahrache, J.: Stabilité et convergence dans les espaces non réflexifs,Sém. d'Anal. Convexe Montpellier 21 (1991), exposé N° 10.
Lahrache, J.: Slice topologie, topologies intermediares, approximées Baire-Wijsman et Moreau-Yosida, et applications aux problèmes d'optimisation convexes,Sém. d'Anal. Convexe Montpellier (1992), exposé N° 3.
Lechicki, A. and Levi, S.: Wijsman convergence in the hyperspace of a metric space,Bull. Un. Mat. Ital. 1-B (1987), 439–452.
Matheron, G.:Random Sets and Integral Geometry, Wiley, New York, 1975.
Michael, E.: Topologies on spaces of subsets,Trans. Amer. Math. Soc. 71 (1951), 152–182.
Mosco, U.: Convergence of convex sets and of solutions of variational inequalities,Advances in Math. 3 (1969), 510–585.
Mosco, U.: On the continuity of the Young-Fenchel transform,J. Math. Anal. Appl. 35 (1971), 518–535.
Naimpally, S.: Wijsman convergence for function spaces,Rend. Circ. Palermo II 18 (1988), 343–358.
Naimpally, S. and Warrack, B.:Proximity Spaces, Cambridge University Press, Cambridge, 1970.
Penot, J.-P.: The cosmic Hausdorff topology, the bounded Hausdorff topology, and continuity of polarity,Proc. Amer. Math. Soc. 113 (1991), 275–286.
Penot, J.-P.: Topologies and convergences on the space of convex functions,Nonlinear Anal. 18 (1992), 905–916.
Poppe, H.: Einige Bemerkungen über den raum der abgeschlossen mengen,Fund. Math. 59 (1966), 159–169.
Phelps, R.:Convex, Functions, Monotone Operators, and Differentiability, Lecture Notes in Math. 1364, Springer-Verlag, Berlin, 1989.
Sonntag, Y.: Convergence au sens de Mosco: théorie et applications à l'approximation des solutions d'inéquations, Thèse. Université de Provence, Marseille, 1982.
Sonntag, Y. and Zalinescu, C.: Set convergences: An attempt of classification, inProc. Intl. Conf. Diff. Equations and Control Theory, Iasi, Romania, August, 1990. Revised version, to appear inTrans. Amer. Math. Soc.
Tsukada, M.: Convergence of best approximations in a smooth Banach space,J. Approx. Theory 40 (1984), 301–309.
Wets, R.J.-B.: Convergence of convex functions, variational inequalities and convex optimization problems, in R. Cottle, F. Gianessi, and J.-L. Lions (eds.),Variational Equations and Complementarity Problems, Wiley, New York, 1980.
Wijsman, R.: Convergence of sequences of convex sets, cones, and functions,Bull. Amer. Math. Soc. 70 (1964), 186–188.
Wijsman, R.: Convergence of sequences of convex sets, cones, and functions, II,Trans. Amer. Math. Soc. 123 (1966), 32–45.
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Beer, G. Wijsman convergence: A survey. Set-Valued Anal 2, 77–94 (1994). https://doi.org/10.1007/BF01027094
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DOI: https://doi.org/10.1007/BF01027094