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Wijsman convergence: A survey. (English) Zbl 0812.54014

The author, a leading authority on hyperspaces, provides an excellent survey of Wijsman convergence which was first introduced thirty years back. Originally meant for convex analysis, the convergence was studied in depth by various workers in hyperspaces. The associated Wijsman hypertopology is a building block of several hypertopologies and provides interesting function space topologies. The paper, which contains an exhaustive bibliography of 70 items, is a must for those interested in hyperspaces.

MSC:

54B20 Hyperspaces in general topology
40A30 Convergence and divergence of series and sequences of functions
52A41 Convex functions and convex programs in convex geometry
46N10 Applications of functional analysis in optimization, convex analysis, mathematical programming, economics
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[1] Attouch, H.:Variational Convergence for Functions and Operators, Pitman, New York, 1984. · Zbl 0561.49012
[2] Attouch, H., Azé, D., and Beer, G.: On some inverse problems for the epigraphical sum,Nonlinear Anal. 16 (1991), 241-254. · Zbl 0743.49005 · doi:10.1016/0362-546X(91)90226-Q
[3] Attouch, H., Lucchetti, R., and Wets, R.: The topology of the ?-Hausdorff distance,Annal. Mat. Pura. Appl. 160 (1991), 303-320. · Zbl 0769.54009 · doi:10.1007/BF01764131
[4] Attouch, H. and Wets, R.: Isometries for the Legendre-Fenchel transform,Trans. Amer. Math. Soc. 296 (1986), 33-60. · Zbl 0607.49009 · doi:10.1090/S0002-9947-1986-0837797-X
[5] Attouch, H. and Wets, R.: Quantitative stability of variational systems: I. The epigraphical distance,Trans. Amer. Math. Soc. 328 (1991), 695-730. · Zbl 0753.49007 · doi:10.2307/2001800
[6] Azé, D.: Caractérisation de la convergence au sens de Mosco en terme d’approximation infconvolutives,Ann. Fac. Sci. Toulouse 8, (1986-1987), 293-314.
[7] Azé, D. and Penot, J.-P.: Operations on convergent families of sets and functions,Optimization 21 (1990), 521-534. · Zbl 0719.49013 · doi:10.1080/02331939008843576
[8] 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.
[9] J.-P. Aubin and Frankowska, H.:Set-Valued Analysis, Birkhäuser, Boston, 1990.
[10] Baronti, M. and Papini, P.-L.: Convergence of sequences of sets, inMethods of Functional Analysis in Approximation Theory, ISNM 76, Birkhäuser-Verlag, Basel, 1986. · Zbl 0606.54006
[11] Beer, G.: Metric spaces with nice closed balls and distance functions for closed sets,Bull. Austral. Math. Soc. 35 (1987), 81-96. · Zbl 0588.54014 · doi:10.1017/S000497270001306X
[12] Beer, G.: An embedding theorem for the Fell topology,Michigan Math. J. 35 (1988), 3-9. · Zbl 0659.54008 · doi:10.1307/mmj/1029003677
[13] Beer, G.: Support and distance functionals for convex sets,Numer. Func. Anal. Optim. 10 (1989), 15-36. · Zbl 0696.46010 · doi:10.1080/01630568908816288
[14] Beer, G.: Convergence of continuous linear functionals and their level sets,Archiv. Math. 52 (1989), 482-491. · Zbl 0671.46005 · doi:10.1007/BF01198356
[15] Beer, G.: Conjugate convex functions and the epi-distance topology,Proc. Amer. Math. Soc. 108 (1990), 117-126. · Zbl 0681.46014 · doi:10.1090/S0002-9939-1990-0982400-8
[16] Beer, G.: Mosco convergence and weak topologies for convex sets and functions,Mathematika 38 (1991), 89-104. · Zbl 0762.46005 · doi:10.1112/S0025579300006471
[17] Beer, G.: A Polish topology for the closed subsets of a Polish space,Proc. Amer. Math. Soc. 113 (1991), 1123-1133. · Zbl 0776.54011 · doi:10.1090/S0002-9939-1991-1065940-6
[18] 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. · Zbl 0824.46089
[19] 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. · Zbl 0786.46006
[20] Beer, G.: Wijsman convergence of convex sets under renorming,Nonlinear Anal. 22 (1994), 207-216. · Zbl 0815.46010 · doi:10.1016/0362-546X(94)90034-5
[21] Beer, G.: Lipschitz regularization and the convergence of convex functions,Numer. Funct. Anal. Optim. 15 (1994), 31-46. · Zbl 0817.49018 · doi:10.1080/01630569408816547
[22] Beer, G. and Borwein, J.: Mosco convergence and reflexicity,Proc. Amer. Math. Soc. 109 (1990), 427-436. · Zbl 0763.46006 · doi:10.1090/S0002-9939-1990-1012924-9
[23] Beer, G.: Mosco and slice convergence of level sets and graphs of linear functionals,J. Math. Anal. Appl. 175 (1993), 53-67. · Zbl 0781.46016 · doi:10.1006/jmaa.1993.1151
[24] Beer, G. and DiConcilio, A.: Uniform convergence on bounded sets and the Attouch-Wets topology,Proc. Amer. Math. Soc. 112 (1991), 235-243. · Zbl 0677.54007 · doi:10.1090/S0002-9939-1991-1033956-1
[25] 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. · Zbl 0774.54004 · doi:10.1007/BF01760016
[26] Beer, G. and Lucchetti, R.: Weak topologies for the closed subsets of a metrizable space,Trans. Amer. Math. Soc. 335 (1993), 805-822. · Zbl 0810.54011 · doi:10.2307/2154406
[27] 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. · Zbl 0812.54015 · doi:10.1216/rmjm/1181072488
[28] Beer, G. and Pai, D.: On convergence of convex sets and relative Chebyshev centers,J. Approx. Theory 62 (1990), 147-169. · Zbl 0733.41030 · doi:10.1016/0021-9045(90)90029-P
[29] Borwein, J. and Fabian, M.: On convex functions having points of Gateaux differentiability which are not points of Frechet differentiability, Preprint. · Zbl 0793.46021
[30] Borwein, J. and Fitzpatrick, S.: Mosco convergence and the Kadec property,Proc. Amer. Math. Soc. 106 (1989), 843-852. · Zbl 0672.46007 · doi:10.1090/S0002-9939-1989-0969313-4
[31] Borwein, J. and Vanderwerff, J.: Dual Kadec-Klee norms and the relationship between Wijsman, slice and Mosco convergence, Preprint, University of Waterloo. · Zbl 0820.46007
[32] Castaing, C. and Valadier, M.:Convex Analysis and Measurable Multifunctions, Lecture Notes in Math. 580, Springer-Verlag, Berlin, 1977. · Zbl 0346.46038
[33] Corradini, P.: Topologie sull’ iperspazio di uno spazio lineare normato, Tesi di laurea, Universitá degli studi di Milano, 1991.
[34] Costantini, C., Levi, S., and Zieminska, J.: Metrics that generate the same hyperspace convergence,Set-Valued Analysis 1 (1993), 141-157. · Zbl 0796.54016 · doi:10.1007/BF01027689
[35] Cornet, B.: Topologies sur les fermés d’un espace métrique, Cahiers de mathématiques de la décision 7309, Université de Paris Dauphine, 1973.
[36] Del Prete, I. and Lignola, B.: On the convergence of closed-valued multifunctions,Boll. Un. Mat. Ital. B 6 (1983), 819-834. · Zbl 0535.54006
[37] Diestel, J.:Geometry of Banach Spaces ? Selected Topics, Lecture Notes in Math. 485, Springer-Verlag, Berlin, 1975. · Zbl 0307.46009
[38] Di Maio, G. and Naimpally, S.: Comparison of hypertopologies,Rend. Istit. Mat. Univ. Trieste 22 (1990), 140-161. · Zbl 0793.54009
[39] Effros, E.: Convergence of closed subsets in a topological space,Proc. Amer. Math. Soc. 16 (1965), 929-931. · Zbl 0139.40403 · doi:10.1090/S0002-9939-1965-0181983-3
[40] Engelking, R.:General Topology, Polish Scientific Publishers, Warsaw, 1977. · Zbl 0373.54002
[41] Fell, J.: A Hausdorff topology for the closed subsets of a locally compact non-Hausdorff space,Proc. Amer. Math. Soc. 13 (1962), 472-476. · Zbl 0106.15801 · doi:10.1090/S0002-9939-1962-0139135-6
[42] 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. · Zbl 0587.54003 · doi:10.1016/0022-247X(85)90246-X
[43] 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.
[44] Hiriart-Urruty, J.-B.: Lipschitzr-continuity of the approximate subdifferential of a convex function,Math. Scand. 47 (1980), 123-134. · Zbl 0426.26005
[45] Holá, L. and Lucchetti, R.: Comparison of hypertopologies, Preprint.
[46] Holmes, R.:Geometric Functional Analysis, Springer-Verlag, New York, 1975. · Zbl 0336.46001
[47] Hörmander, L.: Sur la fonction d’appui des ensembles convexes dans une espace localement convexe,Arkiv. Mat. 3 (1954), 181-186. · Zbl 0064.10504 · doi:10.1007/BF02589354
[48] 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. · Zbl 0282.46005
[49] Klein, E. and Thompson, A.:Theory of Correspondences, Wiley, New York, 1984. · Zbl 0556.28012
[50] Kuratowski, K.:Topology, vol. 1, Academic Press, New York, 1966.
[51] M. Lavie: Contribution a l’étude de la convergence de sommes d’ensembles aléatoires indépendants et martingales multivoques, Thèse, Montpellier, 1990.
[52] Lahrache, J.: Stabilité et convergence dans les espaces non réflexifs,Sém. d’Anal. Convexe Montpellier 21 (1991), exposé N{\(\deg\)} 10.
[53] 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{\(\deg\)} 3. · Zbl 1267.49015
[54] Lechicki, A. and Levi, S.: Wijsman convergence in the hyperspace of a metric space,Bull. Un. Mat. Ital. 1-B (1987), 439-452. · Zbl 0655.54007
[55] Matheron, G.:Random Sets and Integral Geometry, Wiley, New York, 1975. · Zbl 0321.60009
[56] Michael, E.: Topologies on spaces of subsets,Trans. Amer. Math. Soc. 71 (1951), 152-182. · Zbl 0043.37902 · doi:10.1090/S0002-9947-1951-0042109-4
[57] Mosco, U.: Convergence of convex sets and of solutions of variational inequalities,Advances in Math. 3 (1969), 510-585. · Zbl 0192.49101 · doi:10.1016/0001-8708(69)90009-7
[58] Mosco, U.: On the continuity of the Young-Fenchel transform,J. Math. Anal. Appl. 35 (1971), 518-535. · Zbl 0253.46086 · doi:10.1016/0022-247X(71)90200-9
[59] Naimpally, S.: Wijsman convergence for function spaces,Rend. Circ. Palermo II 18 (1988), 343-358. · Zbl 0649.54008
[60] Naimpally, S. and Warrack, B.:Proximity Spaces, Cambridge University Press, Cambridge, 1970. · Zbl 0206.24601
[61] Penot, J.-P.: The cosmic Hausdorff topology, the bounded Hausdorff topology, and continuity of polarity,Proc. Amer. Math. Soc. 113 (1991), 275-286. · Zbl 0774.54008 · doi:10.1090/S0002-9939-1991-1068129-X
[62] Penot, J.-P.: Topologies and convergences on the space of convex functions,Nonlinear Anal. 18 (1992), 905-916. · Zbl 0797.46008 · doi:10.1016/0362-546X(92)90128-2
[63] Poppe, H.: Einige Bemerkungen über den raum der abgeschlossen mengen,Fund. Math. 59 (1966), 159-169. · Zbl 0139.40404
[64] Phelps, R.:Convex, Functions, Monotone Operators, and Differentiability, Lecture Notes in Math. 1364, Springer-Verlag, Berlin, 1989. · Zbl 0658.46035
[65] Sonntag, Y.: Convergence au sens de Mosco: théorie et applications à l’approximation des solutions d’inéquations, Thèse. Université de Provence, Marseille, 1982.
[66] 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.
[67] Tsukada, M.: Convergence of best approximations in a smooth Banach space,J. Approx. Theory 40 (1984), 301-309. · Zbl 0545.41042 · doi:10.1016/0021-9045(84)90003-0
[68] 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. · Zbl 0481.90066
[69] Wijsman, R.: Convergence of sequences of convex sets, cones, and functions,Bull. Amer. Math. Soc. 70 (1964), 186-188. · Zbl 0121.39001 · doi:10.1090/S0002-9904-1964-11072-7
[70] Wijsman, R.: Convergence of sequences of convex sets, cones, and functions, II,Trans. Amer. Math. Soc. 123 (1966), 32-45. · Zbl 0146.18204 · doi:10.1090/S0002-9947-1966-0196599-8
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