Notes and References
Chapter 1
The material in § 1.1 is very well-known. There are many accessible accounts of general topology; for our purposes, we recommend the monographs of Dugundji [Dug] and Willard [Wil]. Somewhat more imposing is the encyclopedic monograph of [En], which gives an unmatched account of the literature. The monograph of Berge [Bg], although omitting many aspects of general topology, presents many of the constructs of set-valued analysis that the reader will encounter in subsequent chapters, and at the same time presents significant material on convex sets and the elements of nonlinear analysis. In some sense, our text seeks to maintain the flavor of [Bg], often walking the border between point-set topology and geometric functional analysis. For the latter subject, we recommend the monographs of Holmes [Holl-2] and Giles [Gil]. The ball measure of noncompactness as introduced in § 1.1 and its relatives are considered in the monograph of Banas and Goebel [BaG]. Kuratowski's Theorem [Ku1] presented in Exercise 1.1.4 has numerous applications; it is a curious fact that for a decreasing sequence
306 NOTES AND REFERENCES 307 convex sets in finite dimensions, we recommend [Lay,MS]. The differentiation of convex functions has been a focal point of research in recent years (see, e.g., [Gil,Ph]). The idea of excess (ecart) originates with Pompieu in 1905 [Porn]. The name "gap functional" was forcefully suggested to the author by Penot. Lemma 1.5.1 can be found in [Cnt], although it was known much earlier for bounded sets.
Chapter 2
In view of the centrality of weak topologies in modern functional analysis, it is a little odd that the the creation of a theory of hyperspaces based on weak topologies was not given adecent try until weIl after 1980. Efforts in this direction have their roots in the study of topologies on lower semicontinuous functions, both convex and nonconvex [Verl-2,Jol], but it was the 1985 paper of Francaviglia, Levi and Lechicki [FLL], who considered distance functionals with a fixed point argument as a function of a set variable, that focused attention on the general method. The notion of strict d-inclusion originates with Costantini, Levi and Zierninska [CLZ]. Paralleling Theorem 2.1.10, necessary and sufficient conditions for Wijsman topologies to agree for closed convex sets have been identified by Corradini [Cor2]. The Vietoris topology was introduced in the early 1920's by Vietoris in [Viel-2J, and its basic topological properties were identified by Michael [Mic 1] in a 1951 paper that set the agenda for the study of hyperspaces for the next twenty years. In terms of later developments, we cite Keesling's deep study of compactness in the hyperspace [Keel-2J, as weIl as Smithson's interesting results on first countability [Sm4]. The term "proximal hit-and-miss topology" originates with Naimpally and his collaborators [BLLN, DCNS], although the proximal topology of §2.2 is implicit in the work of Nachmann [Nac]. Sequential convergence with respect to the proximal topology was studied by Fisher [Fis] and is sometimes called Fisher convergence in the literature (see, e.g., [BaP]). A number of authors had tried to pin down the precise relationship between the Wijsman topology, the ball proximal topology, and the topology of Exercise 2.2.12 (see, e.g., [FLL,Be8, DMN,BTa1] before Hola and Lucchetti [HL] clarified the issue, using the constructs of [CLZ]. Hit-and-miss topologies in the abstract were studied by Poppe [Popl-2] in papers that should have attracted more attention than they received. Given a farnily of weak topologies on a set, the supremum of these topologies is again a weak topology, as determined by the amalgamation of the defining functionals. In particular, suprema of hyperspace topologies can often be attractively expressed as weak topologies. Clean presentations of infima of hyperspace topologies are generally difficult to produce. One problem is a lack of distributivity within the lattice of hyperspace topologies. With distributivity, the infimum of the Vietoris topology and the Hausdorff metric topology ought to be the proximal topology. But as Levi, Lucchetti, and Pelant have shown, this occurs if and only if the underlying metric space is either UC or totally bounded [LLP]. UC spaces perhaps originated with Nagata [Nag] and are periodically rediscovered every few years. It is of course possible to consider UC spaces in the context of uniform spaces [Ats2,NS]. Theorem 2.3.4 is a variant of a result of the author [Be4], who used the Hausdorff metric topology in lieu of the proximal topology. For a general discussion of hyperspaces and function spaces, consult [Nai3]. The slice topology was anticipated by a hit-and-miss topology compatible with Mosco convergence of sequences of convex sets [Be 10-11]. The topology itself was first introduced by Sonntag and Zalinescu [SZ2] as a weak topology determined by a farnily of gap functionals (see Corollary 4.l.7) as part of an overall program to classify hyperspace topologies. Around the same time, the author [Be19-20] introduced the slice topology in 308 NOTES AND REFERENCES much the same way in an attempt to come up with a topology a good deal stronger than the Wijsman topology that remained compatible with the measurability of convex-valued multifunctions (see §6.5). For lower semicontinuous convex functions identified with their epigraphs, the topology agrees with a topology of Joly [Jol] defined in terms of the lower semicontinuity of epigraphieal multifunctions, as we show in §8.2 The author does not know whether the Wijsman topology corresponding to an arbitrary compatible metrie for a Polish space is necessarily Polish. This is true in the special case that the metrie d is totally bounded, as established by Effros [Eff]. The author is hopeful that more transparent proofs of the results of §2.5 can be found, as weH as a useful complete metrie for the Wijsman topology for the closed subsets of a Polish space.
Chapter 3
Attouch-Wets convergence appears in Mosco's initial 1969 paper on Mosco convergence [Mos1], but was not studied in depth until many years later by Attouch and Wets [AW1-2]. Theorem 3.1.3 was first proved by Attouch, Lucchetti, and Wets [ALW], modifying the standard proof of completeness for the Hausdorff metric topology, as established for bounded sets by Hahn [Hah2]. The Hausdorff metric topology originates with Pompieu [Pom] , but not the Hausdorff metrie itself, as Pompieu worked with the equivalent metric p(A, B) =ecJCA,B) + ed(B,A). Without question, the Hausdorff metrie topology is the most weH-studied hyperspace topology, and §3.2 merely scratches the surface. A great deal of attention has been focused on the Hausdorff metric applied to metric continua [Nad]. A central result in this theory is the Curtis-Schiori Theorem [CS]: the space of closed subsets of a locally connected metrie continuum is homeomorphic to the Hilbert cube. An equal amount of attention has been given to the space of closed and bounded convex sets in finite dimensions, equipped with the Hausdorff distance. In this direction, curious Baire category results have been obtained, especially by Zarnfirescu (see, e.g., [Gru]). As another result within this framework, Vitali [Vit2] has obtained a Hörmander• type theorem replacing uniform distance between support functions by LP-distances, computed through integration on the surface of the unit ball equipped with normalized Lebesgue measure. Moreover, such a distance is bounded below by an expression involving the uniform distance, Le., ordinary Hausdorff distance. The Bulgarian school of approximation under the leadership of Sendov [Sen] has developed a theory of constructive approximation of functions based on Hausdorff distance. This theory not only produces analogues of the classieal theory in the continuous case, but also comfortably handles discontinuous bounded real functions by identifying such functions with the set of points lying between the lower and upper envelopes of the function (see §6.1). Hausdorff distance naturally arises in the theory of fractals [Hut,Wie]. There is also literature on descriptive set theoretic questions involving Hausdorff distance. The Hausdorff uniform topology was studied along with the Vietoris topology by Michael [Micl], and many facts cited in this monograph about both topologies come from this source. It has been shown by Ward [War] that distinct uniformities may determine the same Hausdorff uniform topology. Theorem 3.3.12 which appeared in [BHPV] was perhaps the first result giving a nontrivial presentation of a supremum of related hyperspace topologies, and influenced the direction of subsequent research in the field. The locally finite topology itself was NOTES AND REFERENCES 309 eonsidered earlier and independently by Marjanovic [Mar] and by Feiehtinger [Fei]. For a reeent variant ofthe locally finite topology, the reader may eonsult [DC]. The equivalenee between strong eonvergenee of linear funetionals and Attoueh• Wets eonvergenee of graphs is anticipated by the notion of "Kato gap" between subspaees [Kat, pp. 197-204].
Chapter 4
Continuity of various set funetionals (diameter, measures of noneompaetness, gap, exeess, . . .) with respeet to Hausdorff distanee has been known for many years, and attempts to prove eontinuity of set funetionals with respeet to weaker topologies are seattered throughout the literature (see, e.g., [BaP,SoD. A general program aimed at the eharaeterization of topologies in terms of the eontinuity of set funetionals was earried out by the author and his assoeiates over the last six years [BelO,Be13,BeP,BLLN,BeL3]. In a similar direetion, Sonntag and Zalineseu [SZ2-3] foeused on classes of uniformities eompatible with hyperspaee topologies, subsuming not only weak topologies but also Cornet's program [Cnt] of embedding CL(X) into C(X,R) , under the identifieation A H d(- ,A). Particular uniforrnities provide the maehinery for quantitative estimation and approximation, as aetively pursued by Attoueh, Wets, Az6, and Penot. One eould easily take as primitive objeets in a study of hyperspaee topology the upper and lower halves of topologies as presented in §4.2 (see, e.g., [FLLD. We mention that the upper Wijsman topology when speeialized to points has been studied in the eontext ofBanaeh spaee geometry by Godefroy and Kalton [GK]. The precise history of the eonneetion between infimal value funetionals and eonvergenee is obseure, and early results in this direetion often appear in multifunetion form [Cho2,DRl]. Attention was foeused on this eonneetion only reeently, again through the work by Sonntag and Zalineseu [SZ2], who euriously did not explicitly bring this within their uniforrnity framework. Their results were eonfined to the metrie setting. We anticipate in the near future topological results involving proximity and eompaetifieation in whieh infimal value funetionals playa pivotal role. Interest in sealar eonvergenee sterns not only from the fundamental 1954 paper of Hörmander [Hö] but also from the expository paper of Salinetti and Wets [SW2]. As a variant of sealar eonvergenee, one ean study pointwise eonvergenee of support funetionals for truneated sets, following Theorem 2.4.8 [BZ].
Chapter 5
The Fell topology was introdueed by Fell in 1962 [Fel] in eonneetion with eertain nonHausdorff but loeally eompaet spaees arising in the theory of C*-algebras. His topology was antieipated by the lbe topology ofMrowka [Mrl]; for both papers, Miehael's article [Miei] is again influential. The notions of upper and lower closed limits of sequenees of sets in §5.2 are unifQrrnly attributed to Painleve, and perhaps first appear in print in articles by his student Zoretti [Zorl-2]. Set eonvergenee of sequenees defined as equality of these limits was subsequently popularized by Hausdorff [Hau] and then by Kuratowski [Ku3]. Convergenee of filters of sets was first eonsidered by Choquet [Chol] in 1947. Convergenee of nets of sets in an equivalent formulation was studied about ten years later by Mrowka [Mrl], who was unaware ofChoquet's work. Shortly thereafter, Kuratowski• Painleve convergence of nets was rediscovered by Frolik [Fro]. Without question, 310 NOTES AND REFERENCES
Choquet's article and the general dissemination of much of its contents through the monograph of Berge [Bg] had a huge influence on the development of the subject, and when convergence is considered in applications, filters rather than nets are traditionally used. In this matter, and despite the equivalence between filter and net convergence (see, e.g., [Wil, § 12]), the present monograph is heretical. Mrowka' s Theorem was earlier established for regular spaces by Frolik [Fro]. The compatibility of Kuratowski-Painleve convergence with the Fell topology in locally compact spaces is a result of Mrowka [Md]; for sequences, this holds in k-spaces [DM2], i.e., spaces in which a set is open if and only if its intersection with each compact subset is relatively open. In relation to Wijsman convergence, Kuratowski-Painleve convergence of nets of nonempty closed sets can be shown to be dual to Vietoris convergence, as expressed by Theorem 2.2.5: in a metrizable space X, A = K-Lim A;., if and only if there exists a compatible metric d such that Chapter 6 This chapter presents the thinnest coverage of the subject matter, and to partially compensate, we provide more elaborate comments here. There is a vast literature on set-valued functions, also called carriers or correspondences. As general references, we recommend the monographs of Berge [Bg], NOTES AND REFERENCES 311 Castaing and Valadier [CV], Klein and Thompson [KT], and Aubin and Frankowska [AuF]. Certainly the most well-studied set-valued function is the metric projection [Holl, Deu,Sin2-3 ,BoFi 1, VI]. Upper and lower semicontinuity as we have presented them originate with Kuratowski [Ku2] and Bouligand [Bou], following earlier consideration of notions of semicontinuity based on lower and upper c10sed limits of sets [Hil,Hur]. Absent an agreed-upon topology for sets, there are many possible formulations of continuity for set• valued functions. For example, between the extreme notions of outer semicontinuity and upper semicontinuity lies Hausdorff upper semicontinuity corresponding to the condition (2) of Lemma 6.2.6. For multifunctions with values in a normed linear space E, another productive intermediate notion is upper hemicontinuity, corresponding to the upper semicontinuity of the composite functions x ~ s(y,T(x)) for each Y E E*, which was implicit in our formulation of the Kakutani Fixed-Point Theorem (see, e.g., [AuF,LR]). Condition (1) ofLemma 6.2.6 is astronger alternative to lower semicontinuity. On the weaker side, the continuity condition of Deutsch and Kenderov [DK] has been particularly useful in the study of metric projections, whereas the slightly stronger condition of Christensen [Chr2] of Exercise 6.6.4, which he called Kenderov continuity, has been exploited by Kenderov, Stegall and their associates in the context of usco maps (see, e.g., [Stel-2, Ken2,ChK,BeK2,BelO]). Weaker still is the notion of ejficiency as formulated by the Veronas [VVl] at a point x: T(w) hits some bounded set for all points w near x. For arecent survey regarding lower semicontinuity concepts, the reader may consult [PR]. A colorful account of the early history of semicontinuous muItifunctions has been written by McAlister [McA]. Our presentation of the KKM principle and its applications borrows heavily from [DuG,Gra,Siml]. The Kakutani fixed point theorem for convex valued usco maps was established in finite dimensions by Kakutani [KakI] in 1941; the extension to normed linear spaces and then to locally convex spaces was achieved by Bohenblust and Karlin [BKa] , Fan [Fant], and Glicksberg [Gli]. It is a straight-forward exercise to go from the finite dimensional version to infinite dimensions, using the proof strategy of Theorem 6.4.7 (see, e.g., [Gli]). For applications to variational inequalities, the reader may consult [DuG,Mos3, Las]. Fan's contributions to the subject are substantial. In §6.5, we considered two approaches to measurability for multifunctions. There is an additional possibility: a set-valued function from Chapter 7 A theory of convexity where sets rather than functions are the primary objects relies on polarity as the primary carrier of duality. In a theory where functions take center stage, Fenchel conjugacy replaces polarity in this role. The former approach seems more geometrical, but the latter is often more convenient for the purposes of analysis. It is possible to develop general conjugacy schemes for nonconvex functions that have been particularly fruitful in the study of quasi-convex functions [GP,Sin4,IN,Crx,VoI2, PV]. For bivariate functions that are convex and l.s.c. in one variable and concave and u.s.c. in the other, one can perform partial conjugations [Roc4]. Epi-hypo convergence of such objects is considered in [A Wl,AAW,Sou2]. Continuity of conjugacy with respect to the Attouch-Wets topology in finite dimensions was established by Wijsman [Wij], in which case the Attouch-Wets topology reduces to the Wijsman topology. The key step in extending this to infinite dimensions is the 1967 Walkup-Wets Isometry Theorem for polar cones, which the authors stated only for reflexive spaces [WW], although their proof with very slight modifications works in complete generality. Their work is the point of departure for a number of subsequent isometry theorems (see, e.g., [AW2,Pe3)), and its simplicity and elegance provides ammunition for those who argue that convex sets (especially cones) rather than functions ought to be the central objects in a discussion of convergence and duality. In this mono graph we focus on the relationship between convergence of convex functions and convergence of their simplest regularizations, namely their Lipschitz NOTES AND REFERENCES 313 regularizations. In the literature, quadratic regularizations, where the smoothing kernel with respect to which epi-sums are formed is of the type JlII·1I2, have received greater attention. Such regularizations may be also be combined with linear perturbations, and often assumptions are made on the norm of the space to guarantee a degree of differentiability of the regularized functions. Two themes run through this work in the literature : (1) regularized functions are at least as well-behaved as their kerneis; (2) convergence of convex functions can be "measured" by the convergence of regularized functions with respect to classical uniformities. There is yet no agreed-upon set of minimal properties for smoothing kerneis. The question of when set convergence of regularizations gives back convergence offunctions is pursued in [AAB]. Well-posed optimization problems are the focus of arecent monograph ofDontchev and Zolezzi [DZ]. Several beautiful generic results of a purely topological nature regarding v well-posedness of continuous functions have been displayed by Coban, Kenderov and Revalski [CKR]. For example, if X is a completely regular Hausdorff space, then most functions in the space Cb(X,R) are well-posed if and only if X contains a dense and completely metrizable subspace. For a curious result linking well-posedness and UC spaces, the reader may consult [RZ]. Chapter 8 V. Klee's review ofWijsman's paper [Math. Reviews 28 (1964) # 514] suggested a simpler path to Wijsman's continuity theorem [Wij] for the Fenchel transform, based on the equivalence in finite dimensions of Wijsman convergence of sequences of nonempty closed sets with their Kuratowski-Painleve convergence. Whether or not Mosco was influenced by Klee's observations, he succeeded 1971 in extending Wijsman's sequential continuity results to reflexive spaces, requiring for convergence Kuratowski-Painleve convergence with respect to both the strong and weak topologies [Mos2]. Not long thereafter, Joly wrote an abstract paper [Jol] on topologies on convex functions defined on a locally convex space, introducing formally the weakest topology on the convex functions such that the epigraphical multifunetions f ~ epi f and f ~ epi f* are lower semieontinuous, subsuming Moseo's results in the process (see also [Bae,Zab]). This often-quoted work is a highly respeeted contribution to the field. But at the same time, it was and is poorly understood. What does Joly's weak topology look like in simple geometrie terms? Conerete geometrie al deseriptions of Joly's topology, reformulated as the slice topology, were given by the author in 1991 [BeI9-20]. But the tools for this analysis were put in plaee several years earlier in [Bell,BeI8], in an investigation ofthe topology of Moseo convergence for convex functions defined on reflexive spaces. In particular, one can find the quantitative separation lemma of §8.2 applied in the arguments therein. Much work remains to be done in the locally convex setting. The Ekeland Principle can be formulated for a complete metric space and actually characterizes completeness for the space [Su]. The Ekeland Principle is an indispensable tool in nonlinear analysis, as documented in [Ek,DF]. It is notable that this result is a relatively modern contribution. A close relative of the Ekeland Principle is the Drop Theorem of Danes [Dan,Penl]. In the convex case, many of the consequences of the Ekeland Principle are anticipated by versions ofthe Bishop-Phelps Theorem, which predate the Ekeland Principle by several years. In particular, Proposition 8.3.6 in set form is a particular manifestation of Bishop-Phelps [BiP]. A strong Ekeland variational principle has been formulated by Georgiev [Geo]. 314 NOTES AND REFERENCES Attoueh's Theorem [Atl] is an historieally eentral result in eonvergenee theory; its extension to nonreflexive spaees in [ABI] depends on Proposition 8.3.6. A very different proof of this subtangent result has been given by Simons [Sim3]. An analogue of Attoueh's Theorem for ordinary Kuratowski-Painleve eonvergenee in a separable reflexive spaee has been given by Matzeu [Mtz]. The Veronas have established aversion of Attoueh's Theorem in noneomplete spaees, formulated in terms of e-subdifferentials [VV2]. In finite dimensions at least, the dass of funetions with respeet to whieh the result is valid ean be usefully enlarged, as observed by Poliquin [PoI2]. At this writing, the validity ofthe natural analogue of Attoueh's Theorem for Attoueh-Wets eonvergenee in a general Banaeh spaee remains to be eonfirmed (but see [ANT]). Stability of the Geometrie Ekeland Theorem for noneonvex funetions was studied by Attoueh and Riahi [AR]; we antieipate that additional results in this more general and less transparent setting will be obtained in the future. Bibliography [AL] P. Alart and B. Lernaire, Penalization in non-classical convex programming via variational convergenee, Math. Programming 51 (1991),307-331. [AmS] A. Ambrosetti and C. Sbordone, r--eonvergenza e G-eonvergenza per probierni non lineari di tipo ellittieo, Boll. Uno Mat.ltal13-A (1976),352-362. [Amr] D. Amir, Best simultaneous approximation (Chebyshev centers), in Inter. Series Num. Math., vol. 72, Birkhäuser-Verlag, Basel, 1984, 19-35. [AnC] H. Antosiewicz and A. Cellina, Continuous selections and differential relations, J. Differential Equations 19 (1975), 386-398. [ArL] V. Arkin and V. Levin, Convexity of values of veetor integrals, theorems on measurable choice, and variational problems, Russian Math. Surveys 27 (1972),21-85. [Ars] Z. Artstein, Set valued measures, Trans. Amer. Math. Soe. 165 (1972), 103- 121. [ArV] Z. Artstein and R. Vitale, A strong law of large numbers for random compaet sets, Ann. Probab. 3 (1975),879-882. [AsR] E. Asplund and R. Roekafellar, Gradients of eonvex functions, Trans. Amer. Math. Soe. 139 (1969),443-467. [Atsi] M. Atsuji, Uniform eontinuity of eontinuous funetions of metric spaces, Pacifie 1. Math. 8 (1958),11-16. [Ats2] ___, Uniform eontinuity of eontinuous funetions on uniform spaees, Canad. 1. Math. 13 (1961), 657-663. [Atl] H. Attouch, Convergenee de fonetions convexes, des sous-differentiels et semi-groupes assoeies, C. R. Aead. Sei. Paris 284 (1977), 539-542. [At2] ___, Familles d'operateurs maximaux monotones et mesurabilite, Ann. Mat. Pura Appl. 120 (1979), 35-111. [At3] ~ __, Variational convergenee for funetions and operators, Pitman, Boston, 1984. [AAB] H. Attoueh, D. Aze and G. Beer, On some inverse stability problems for the epigraphical sum, Nonlinear Anal. 16 (1991), 241-254. [AAW] H. Attoueh, D. Aze, and R. Wets, On eontinuity properties of the partial Legendre-Fenchel trans form : eonvergence of sequenees of augmented Lagrangian funetions, Moreau-Yosida approximates, and subdifferential operators, in Fermat Days 85: Mathematies for optimization, J.-B. Hiriart• Urruty ed., Elsevier North-Holland, 1986. [ABI] H. Attouch and G. Beer, On the eonvergence of subdifferentials of eonvex functions, Sem. d'Anal. Convexe Montpellier 21 (1991), expose N° 8; Areh. Mat. 60 (1993), 389-400 . [AB2] ...,.,.-:_-:--:' Stability of the geometrie Ekeland variational principle, Sem. d'Anal. Convexe Montpellier 21 (1991), expose N° 9; to appear, J. Optim. Theory Appl. [ALW] H. Attoueh, R. Luechetti, and R. Wets, The topology of the p -Hausdorff distance, Ann. Mat. Pura Appl. 160 (1991), 303-320, [ANT] H. Attoueh, J. Ndoutoume, and M. Thera, Epigraphical convergence of functions and convergence of their derivatives in Banaeh spaees, Sem. d'Anal. Convexe Montpellier 20 (1990), expose W 9. [AR] H. Attoueh and H. Riahi, Stability results for Ekeland's c-variational principle and eone extremal solutions, Math. Oper. Res. 18 (1993), 173-201. [AWl] H. Attoueh and R. Wets, A eonvergence theory for saddle funetions, Trans. Amer. Math. Soc. 280 (1983),1-41. [AW2] ___,Isometries for the Legendre-Fenchel transform, Trans. Amer. Math. SOC. 296 (1986), 33-60. 315 316 BIBLIOGRAPHY [AW3] -,-__, Quantitative stability of variational systems: I. The epigraphical distance, Trans. Amer. Math. Soc. 328 (1991), 695-730. [AW4] _-::-:-_, Quantitative stability of variational systems : 11. A framework for nonlinear conditioning, SIAM 1. Optimization 3 (1993), 359-381. [Au] J.-P. Aubin, Contingent derivatives of set-valued maps and existence of solutions to nonlinear inclusions and differential inclusions, Adv. in Math., Supplementary Studies, L. Nachbin, ed., 1981, 160-232. [AuC] J.-P. Aubin and A. Cellina, Differential inclusions, Springer-Verlag, Berlin, 1984. [AuE] J.-P. Aubin and I. Ekeland, Applied Nonlinear Analysis, Wiley, New York, 1984 [AuF] J.-P. Aubin and H. Frankowska, Set-Valued Analysis, Birkhäuser, Boston, 1990. [Aumi] R. Aumann, Integrals of set-valued functions, J. Math. Anal. Appl. 12 (1965), 1-12. [Aum2] ::--_-::-' Measurable utility and the measurable choice theorem, La Decision, Proc. Int. Coloq., Aix-en-Provence (1967), p. 15-26. [Az] D. Aze, Caracterisation de la convergence au sens de Mosco en terme d'approximation inf-convolutive, Ann. Fac. Sci. Toulouse 8 (1986-1987), 293-314. [AP1] D. Aze and J.-P. Penot, Operations on convergent families of sets and functions, Optirnization 21 (1990), 521-534. [AP2] =----:_, Qualitative results about the convergence of convex sets and convex functions, in Optirnization and nonlinear analysis (Haifa, 1990), Res. Notes Math. Sero #244, Longman Sci. Tech., Harlow, 1992, 1-24. [Bac] K. Back, Continuity of the Fenchel trans form of convex functions, Proc. Amer. Math. Soc. 97 (1986), 661-667. [Bai] R. Baire, These. Ann. Mat. Pura Appl. 3 (1899). [BZ] A. Balayadi and C. Zalinescu, Bounded scalar convergence, preprint. [BaI] E. Balder, A general approach to lower sernicontinuity and lower closure in optimal control theory, SIAM 1. Control Opt. 22 (1984), 570-598. [BaG] J. Banas and K. Goebel, Measures of noncompactness in Banach spaces, Lecture notes in pure and applied mathematics #60, Decker, New York, 1980. [BJ] H. T. Banks and M. Jacobs, A differential calculus for multifunctions, J. Math. Anal. Appl. 29 (1970), 246-272. [BaP] M. Baronti and P.-L. Papini, Convergence of sequences of sets, in Functional analysis and approximation theory, Bombay, 1985, in Inter. Series Num. Math. vol. 76, Birkhäuser, Basel, 1986. [BrG] R. Bartle and L. Graves, Mappings between function spaces, Trans. Amer. Math. Soc. 72 (1952),400-413. [Bel] G. Beer, Upper semicontinuous functions and the Stone approximation theorem, J. Approx. Theory 34 (1982), 1-11 . [Be2] ..,..,..,....,..".,---' On functions that approximate relations, Proc. Amer. Math. Soc. 88 (1983), 643-647. [Be3] ___, More on convergence of continuous functions and topological convergence of sets, Canad. Math. BuH. 28 (1985), 52-59. [Be4] _-:-_, Metric spaces on which continuous functions are uniformly continuous and Hausdorff distance, Proc. Amer. Math. Soc. 95 (1985), 653- 658. [Be5] ---c:-::----::-' More about metric spaces on which continuous functions are uniformly continuous, Bull. Austral. Math. Soc. 33 (1986), 397-406. [Be6] ___, Lattice sernicontinuous mappings and their application, Houston 1. Math. 13 (1987),303-318. [Be7] ___, On a measure of noncompactness, Rend. Mat. 7 (1987), 193-205. [Be8] __.,--' Metric spaces with nice closed balls and distance functions for closed sets, BuH. Austral. Math. Soc. 35 (1987), 81-96. BIBLIOGRAPHY 317 [Be9] ___" UC spaces revisited, Amer. Math. Monthly. 95 (1988), 737-739. [Be10] ___, On Mosco convergence of convex sets, Bull. Austral. Math. Soc. 38 (1988), 239-253. [Bell] ___, On the Young-Fenchel transform for convex functions, Proc. Amer. Math. Soc. 104 (1988),1115-1123. [Be12] ___, An embedding theorem for the Fell topology, Michigan Math. J. 35 (1988), 3-9. [Be 13] ___, Support and distance functionals for convex sets, Numer. Funct. Anal. Optim. 10 (1989), 15-36. [Be14] -::-:-:-::--=-::-" Infima of convex functions, Trans. Amer. Math. Soc. 315 (1989), 849-859. [Be15] =....,..,..,::-::-::' Convergence of linear functionals and their level sets, Arch. Math. 55 (1990), 285-292. [Be16] -,--_-:-::' Conjugate convex functions and the epi-distance topology, Proc. Amer. Math. Soc. 108 (1990),117-126. [Be17] ___, A Polish topology for the closed subsets of a Polish space, Proc. Amer. Math. Soc. 113 (1991),1123-1133. [Be18] -::----:-_' Mosco convergence and weak topologies for convex sets and functions, Mathematika 38 (1991), 89-104. [Be19] ____,Topologies on closed and closed convex sets and the Effros measurability of set valued functions, Sem. d' Anal. Convexe Montpellier 21 (1991), expose W 2. [Be20] __-::-' The slice topology: A viable alternative to Mosco convergence in nonreflexive spaces, Sem. d'Anal. Convexe Montpellier 21 (1991), expose N° 3; Nonlinear Anal. 19 (1992), 271-290. [Be21] __,--,' Topological completeness of function spaces arising in the Hausdorff approximation of functions, Canad. Math. Bull. 35 (1992), 439- 448. [Be22] ___, On the Fell topology, Set-Valued Anal. 1 (1993),69-80. [Be23] ___, Wijsman convergence of convex sets under renorming, to appear, Nonlinear Anal. [Be24] _--:-_, Lipschitz regularization and the convergence of convex functions, preprint. [Be25] -=--=-_' A note on epi-convergence, to appear, Canad. Math. Bull. [BB1] G. Beer and J. Borwein, Mosco convergence and reflexivity, Proc. Amer. Math. Soc. 109 (1990), 427-436. [BB2] -::----:-_' Mosco and slice convergence of level sets and graphs of linear functionals, J. Math. Anal. Appl. 175 (1993), 53-67. [BeDC] G. Beer and A. DiConcilio, Uniform continuity on bounded sets and the Attouch-Wets topology, Proc. Amer. Math. Soc. 112 (1991), 235-243. [BHPV] G. Beer, C. Himmelberg, C. Prikry, and F. Van Vleck, The locally finite topologyon 2x, Proc. Amer. Math. Soc. 101 (1987), 168-171 . [BeK1] G. Beer and P. Kenderov, On the argmin multifunction for lower semicontinuous functions, Proc. Amer. Math. Soc. 102 (1988),107-113. [BeK2] ---:~,----' Epiconvergence and Baire category, Boll. Uno Mat. Ital. (7) 3-B (1989), 41-56. [BLLN] G. Beer, A. Lechicki, S. Levi, and S. Naimpally, Distance functionals and suprema of hyperspace topologies, Ann. Mat. Pura Appl. 162 (1992), 367- 381. [BeLl] G. Beer and R. Lucchetti, Convex optimization and the epi-distance topology, Trans. Amer. Math. Soc. 327 (1991), 795-813. [BeL2] _-::-:----:-' The epi-distance topology : continuity and stability results with applications to convex optimization problems, Math. Oper. Res. 17 (1992), 715-726. [BeL3] -:----:-c' Weak topologies on the closed subsets of a metrizable space, Trans. Amer. Math. SOC. 335 (1993), 805-822. 318 BIBLIOGRAPHY [BeP] G. Beer and D. Pai, On convergence of convex sets and relative Chebyshev centers, J. Approx. Theory 62 (1990), 147-169. [BRW] G. Beer, R. T. Rockafellar, and R. Wets, A characterization of epi• convergence in terms of convergence of level sets, Proc. Amer. Math. Soc. 116 (1992), 753-761 . [BTal] G. Beer and R. Tamaki, On hit-and-miss hyperspace topologies, to appear, Comm. Math. Univ. Carolin. [BTa2] :--__' The intimal value functional and the uniformization of hit-and-miss hyperspace topologies, preprint. [BTh] G. Beer and M. Thera, Attouch-Wets convergence and a differential operator for convex functions, to appear, Proc. Amer. Math. Soc. [BDMN] A. Bella, G. Di Maio, and S. Naimpally, Abstract measure of farness and hypertopologies, in Proc. Fifth Ital. Inter. Topology Conf., Lecce-Otranto, 1992, 329-337. [Bg] C. Berge, Topological spaces, Macmillan, New York, 1963. [BiP] E. Bishop and R. Phelps, The support functionals of a convex set, in Proc. Symp. Pure Math. 7, Amer. Math. Soc., Providence, Rhode Island, 1963,26- 36. [Bis] J. Bismut, Integrals convexes et probabilites, J. Math. Anal. Appl42 (1973), 639-673. [BKa] H. Bohenblust and S. Karlin, On a thoerem of Ville, in Contributions to the theory of games, vol. 1, Princeton, 1950. [Borl] J. Borwein, A note on e-subgradients and maximal monotonicity, Pacitic J. Math. 103 (1982), 307-314. [Bor2] .,--,-__, Adjoint process duality, Math. Oper. Res. 8 (1983),403-434. [BoFa] J. Borwein and M, Fabian, On convex functions having points of Gateaux differentiability which are not points of Frechet differentiability, preprint. [BoFil] J. Borwein and S. Fitzpatrick, Existence of nearest points in Banach spaces, Canad. J. Math. 41 (1989),702-720. [BoFi2] -=-_~, Mosco convergence and the Kadec property, Proc. Amer. Math. Soc. 106 (1989), 843-852. [BoL] J. Borwein and A. Lewis, Convergence of decreasing sequences of convex sets in nonreflexive Banach spaces, preprint. [BoS] J. Borwein and H. Strojwas, Proximal analysis and boundaries of c10sed sets in Banach space, part I: Theory, Canad. J. Math. 38 (1986), 431-452. [BoV] J. Borwein and J. Vanderwerff, Dual Kadec-Klee norms and the relationship between Wijsman, slice and Mosco convergence, preprint. [BPP] M. Bougeard, J.-P. Penot and A. Pommelet, Towards minimal assumptions for the intimal convolution regularization, J. Approx.Theory 64 (1991), 245- 270. [Bou] G. Bouligand, Sur la semi-continuite d'inc1usions et quelques sujets connexes, Enseignement Mathematique 31 (1932), 14-22. [Bre] H. Brezis, Operateurs maximaux monotones et semi-groupes de contraction dans les espaces de Hilbert, North-Holland, Amsterdam, 1973. [BR] A. BrjZlnsted and R. T. Rockafellar, On the subdifferentiablity of convex functions, Proc. Amer. Math. Soc. 16 (1965), 605-611. [Brol] F. Browder, The tixed point theory of multi-valued mappings in topological vector spaces, Math. Ann. 177 (1968), 283-301. [Br02] __----" Nonlinear operators and nonlinear equations of evolution in Banach spaces, Amer. Math. Soc. Proc. Symp. #18, Providence, Rhode Island, 1976. [Bu] G. Buttazzo, Su una defmizione generale dei T-limiti, Boll. Uno Mat.ltal. 14- B (1977), 722-744. [BDM] G. Buttazzo and G. Dal Maso, T-convergence and optimal control problems, J. Optim. Theory Appl. 38 (1982), 385-407. [Ca] C. Castaing, Sur les multi-applications measurables. These, Caen, 1967. [CV] C. Castaing and M. Valadier, Convex analysis and measurable multifunctions, LNM # 580, Springer-Verlag, Berlin, 1977. BIBLIOGRAPHY 319 [Cell] A. Cellina, Approximation of set-valued funetions and fixed point theorems, Ann. Mat. Pura. Appl. 82 (1969), 17-24. [Ce12] -,---_:--:-' A further result on the approximation of set-valued mappings, Atti Aeead. Naz. Lineei Rend. Cl. Sei. Mat. Natur. 48 (1970),412-416. [Ce13] ___, Multivalued differential equations and ordinary differential equations, SIAM J. Appl. Math. 18 (1970), 533-538. [CeI4] ___, A theorem of non-empty interseetion, Bull. Aead. Polon. des Sei. 20 (1972), 471-472. [Chi] F. Chimenti, On the sequential eompaetness of the spaee of subsets, Bull. Aead. Polon. des Sei. 20 (1972), 221-226. [Cho1] G. Choquet, Convergenees, Ann. Univ. Grenoble 23 (1947-1948),57-112. [Ch02] G. Choquet, Outils topologiques et metriques de l'analyse mathematique, Centre de doeumentation universitaire et S.E.D.E.S reunis, Paris, 1966. [Chrl] J. P. R. Christensen, Topology and Borel strueture, North-Holland, Amsterdam, 1974. [Chr2] _-,-_' Theorems of I. Namioka and R. E. Johnson type for upper sernieontinuous and eompaet-valued set-valued mappings, Proe. Amer. Math. Soe. 86 (1982), 649-655. [ChK] J. P. R. Christensen and P. Kenderov, Dense strong eontinuity of mappings and the Radon-Nikodym property, Math. Seand. 54 (1984), 70-78. [Cio] I. Cioraneseu, Geometry of Banach spaees, duality mappings, and nonlinear problems, Kluwer, Dordreeht, Holland, 1990. [Cl] F. Clarke, Optirnization and nonsmooth analysis, Wiley, New York, 1982. v v [Cobi] M. Coban, Many-valued mappings and Borel sets, I, Trans. Moseow Math. Soe. 22 (1970), 258-280. v [Cob2] -=---=:-=-' Many-valued mappings and Borel sets, II, Trans. Moseow Math. Soe. 23 (1970), 286-310. v [CKR] M. Coban, P. Kenderov, and J. Revalski, Generie well-posedness of optirnization problems in topologieal spaees, Mathematika 36 (1989), 301-324. [Coh] D. Cohn, Measure theory, Birkhäuser, Boston, 1980. [CP] L. Contes se and J.-P. Penot, Continuity of the Fenchel eorrespondenee and eontinuity of polarities, J. Math. Anal. Appl. 156 (1991), 305-328. [Cnt] B. Cornet, Topologies sur les fermes d'un espaee metrique, Cahiers de mathematiques de la decision #7309, Universite de Paris Dauphine, 1973. [Corl] P. Corradini, Topologie suB' iperspazio di uno spazio lineare normato. Tesi di Laurea, Universita degli studi di Milano, 1991. [Cor2] __---:::' Norms that generate the same Wijsman topology on eonvex sets, to appear, Istituto Lombardo Rend. Sei. [CLP] C. Costantini, S. Levi and J. Pelant, Infima of hyperspace eonvergenees, preprint. [CLZ] C. Costantini, S. Levi and J. Zieminska, Metries that generate the same hyperspace eonvergenee, to appear, Set-Valued Anal. [Crx] J.-P. Crouziex, Contributions a l'etude des fonetions quasieonvexes. These, Universite de Clermont-Ferrand II, 1977. [CS] D. Curtis and R. Sehiori, 2x and C(X) are homeomorphie to the Hilbert eube, Bull. Amer. Math. Soe. 80 (1974), 927-931. [DM 1] D. Dal Maso, Questioni di topologia legate alla r-eonvergenza, Rieerehe Mat. 32 (1983), 135-162. [DM2] G. Dal Maso, An introduction to r-eonvergenee, Birkhäuser, Boston, 1993. [DMDG] G. Dal Maso and E. De Giorgi, r-eonvergenee and the ea1culus ofvariations, in Mathematieal theories of optirnization, LNM # 979, Springer-Verlag, Berlin, 1983, 121-143. [Dan] J. Danes, A geometrie theorem useful in nonlinear funetional analysis, Boll. Uno Mat. HaI. 6 (1972), 369-375. 320 BIBLIOGRAPHY [DFS] G. Dantzig, J. Folkman and N. Shapiro, On the continuity ofthe minimum set of a continuous function, J. Math. Anal. Appl. 17 (1967), 519-548. [DB] F. Oe Blasi, The differentiability of multifunctions, Pacific. J. Math. 66 (1976), 67-81. [DBL] F. Oe Blasi and F. Lasota, Characterization of the integral of set-valued functions, Atti Accad. Naz. Lincei Rend. Cl. Sci. Mat. Natur. 46 (1969), 154- 157. [DBMl] F. Oe Blasi and J. Myjak, On continuous approximations for multifunctions, Pacific J. Math. 123 (1986),9-31. [DBM2] , Weak convergence of convex sets in Banach spaces, Arch. Math. 47 (1986),448-456. [Deb] G. Debreu, Integration of correspondences, in Proc. Fifth Berkeley Symp. on Stat. and Prob. vol. 2 (1966), University of Califomia Press, 351-372. [DF] D. G. DeFigueiredo, The Ekeland principle with applications and detours, Tata Institute ofFundamental Research, Springer-Verlag, Berlin, 1989. [001] E. DeGiorgi, Convergence problems for functionals and operators, in Recent methods in nonlinear analysis, E. Oe Giorgi, E. Magenes and U. Mosco, eds., Pitagora Editrice, Bologna, 1979, 131-188. [DG2] , Generalized limits in calculus of variations, in Topics in functional analysis, 1980-1981, F. Strocchi, E. Zarantonello, E. DeGiorgi, G. Dal Maso and L. Modica, eds., Scuola Normale, Pisa, 1981. [DGF] E. DeGiorgi and T. Franzoni, Su un tipo di convergenza variazionale, Atti Accad. Naz. Lincei Rend. Cl. Sci. Mat. Natur. 58 (1975), 842-850. [DPL] I. DeI Prete and B. Lignola, On convergence of closed valued multifunctions, Boll. Uno Mat. Ital. B 6 (1983), 819-834. [DPDL] I. DeI Prete, S. Dolecki, and B. Lignola, Continuous convergence and preservation of convergence of sets, J. Math. Anal. Appl. 120 (1986), 454- 471. [Deu] F. Deutsch, A survey of metric selections, in Contemporary Math. 18 (1983), 49-71. [DIS] F. Deutsch, V. Indumathi, and K. Schnatz, Lower semicontinuity, almost lower semicontinuity, and continuous selections for set-valued mappings, J. Approx. Theory 53 (1988),266-294. [DK] F. Deutsch and P. Kenderov, Continuous selections and approximate selections for set-valued mappings and applications to metric projections, SIAM J. Math. Anal. 14 (1983), 185-194. [DVGZ] R. Deville, G. Godefroy, and V. Zizler, Renormings in Banach spaces, Longman, Harlow, United Kingdom, 1993. [DC] A. Di Concilio, Uniformities, hyperspaces and normality, Monatsh. Math. 107 (1989), 303-308. [DCN] A. Di Concilio and S. Naimpally, Atsuji spaces: continuity versus uniform continuity, in Proc. Sixth Brazilian Topology Meeting, Campinas, Brazil, 1988. [DCNS] A. Di Concilio, S. Naimpally, and P. Sharma, Proximal hypertopologies, in Proc. Sixth Brazilian Topology Meeting, Campinas, Brazil, 1988. [Die] J. Diestel, Geometry ofBanach spaces, LNM #485, Springer-Verlag, Berlin, 1975. [Ddn] J. Dieudonne, Une generalization des espaces compacts, J. Math. Pures Appl. 23 (1944),65-76. [DMN] G. Di Maio and S. Naimpally, Comparison of hypertopologies, Rend. Istit. Mat. Univ. Trieste 22 (1990), 140-161. [Doll] S. Dolecki, Tangency and differentiation: some applications of convergence theory, Ann. Mat. Pura Appl. 130 (1982), 223-255. [Do12] , Convergence of minima in convergence spaces, Optimization 17 (1986), 553-572. [DGL] S. Dolecki, G. Greco and A. Lechicki, When do the upper Kuratowski and the co-compact topologies coincide?, preprint. BIBLIOGRAPHY 321 [DGLM] S. Dolecki, 1. Guillerme, M. B. Lignola, and C. Malivert, r -inequalitites and stability of generalized extremal convolutions, Math. Nachr. 152 (1991), 299- 323. [DL] S. Dolecki and A. Lechicki, On the structure ofupper semicontinuity, J. Math. Anal. Appl. 88 (1982), 547-554. [DR1] S. Dolecki and S. Rolewicz, A characterization of semicontinuity-preserving multifunctions, J. Math. Anal. Appl. 65 (1978),26-31. [DR2] --:--:---=' Metric characterizations of upper semicontinuity, J. Math. Anal. Appl. 69 (1979), 146-152. [DSW] S. Dolecki, G. Salinetti, and R. Wets, Convergence of functions: equi• semicontinuity, Trans. Amer. Math. Soc. 276 (1983), 409-429. [DZ] A. Dontchev and T. Zolezzi, Well-posed optimization problems, LNM # 1543, Springer-Verlag, Berlin, 1993. [Dor] C. Dorsett, Connectivity properties in hyperspaces and product spaces, Fund. Math. 121 (1984), 189-197. [DrL] L. Drewnowski and I. Labuda, On minimal upper semicontinuous compact• valued maps, Rocky Mountain J. Math. 20 (1990), 1-16. [Dug] J. Dugundji, Topology, Allyn and Bacon, Boston, 1966. [DuG] 1. Dugundgi and A. Granas, Fixed Point Theory, Polish Seientific Publishers, Warsaw, 1981. [DS] N. Dunford and J. Schwarz, Linear operators, Part 1, Wiley, New York, 1988. [Eft] E. Effros, Convergence of closed sub sets in a topological space, Proc. Amer. Math. Soc. 16 (1965), 929-931. [ES] N. Efimov and S. Stechkin, Approximative compactness and Chebyshev sets. Soviet Math. Dokl. 2 (1961),1226-1228. [Efr] V. Efremovic, The geometry of proximity I, Math. Sbornik 31 (1952), 189- 200. [Ek] 1. Ekeland, Nonconvex minimization problems, Bull. Amer. Math. Soc. 1 (1979),443-474. [En] R. Engelking, General topology, Polish Scientific Publishers, W arsaw, 1977. [Fan 1] K. Fan, Fixed-point and minimax theorems in locally convex topologicallinear spaces, Proc. Nat. Acad. Sei. 38 (1952), 121-126. [Fan2] __--" A generalization of Tychonoffs fixed point theorem, Math. Ann. 142 (1961), 305-310. [Fan3] :c-=---:--, Applications of a theorem conceming sets with convex sections, Math. Ann. 163 (1966), 189-203. [Fan4] ::-:-::-:-_' A minimax inequality and applications, in Inequalities III, O. Shisha, ed., Academic Press, New York, 1972, 103-113. [Fei] O. Feichtinger, Properties of the ;. topology, in Set-valued mappings, selections, and topological properties of 2x, in Proc. Conf. SUNY Buffalo 1969, W. Fleischman, ed., LNM #171, Springer-Verlag, Berlin, 1970, 17- 23. [Fel] 1. Fell, A Hausdorff topology for the closed subsets of a locally compact non• Hausdorff space, Proc. Amer. Math. Soc. 13 (1962),472-476. [Fen] W. Fenchel, Convex cones, sets, and functions, mimeographed lecture notes, Princeton University, 1951. [Fil] A. Filippov, The existence of solutions of generalized differential equations, Mat. Zemetki 10 (1972),307-313. [Fis] B. Fisher, A reformulation of the Hausdorff metric, Rostock Math. Kolloq. 24 (1983),71-76. [FL] S. Fitzpatrick and A. Lewis, Weak* convergence of convex sets, preprint. [FP] S. Fitzpatrick and R. Phelps, Bounded approximants to monotone operators on Banach spaces, Ann. Inst. H. Poincare Anal. Non Lineaire 9 (1992), 573- 595. 322 BffiLiOGRAPHY [FIa] J. Flachsmeyer, Verschiedene Topologisierungen im Raum der abgeschlossenen Teilmengen, Math. Nachr. 26 (1964), 321-337. [FIe] W. FIeischman, Set-valued mappings, selections and topological properties of 2x, in Proc. Conf. SUNY Buffalo 1969, W. Fleischman, ed., LNM #171, Springer-Verlag, New York, 1970. [Flo] K. Floret, Weakly compact sets, LNM # 801, Springer-Verlag, Berlin, 1980. [Forl] M. K. Fort, Points of continuity of semi-continuous functions, Publ. Math. Debrecen 2 (1951-1952), 100-102. [For2] ___, Category theorems, Fund. Math. 42 (1955), 276-288. [Fr] A. Fougeres and A. Truffert, Regularisation S.C.I. et T-convergence approximations inf-convolutives associees a un referential, Ann. Mat. Pura Appl. 152 (1988), 21-51. [FLL] S. Francaviglia, A. Lechicki, and S. Levi, Quasi-uniformization of hyperspaces and convergence of nets of semicontinuous multifunctions, J. Math. Anal. Appl. 112 (1985), 347-370. [Fra] J. Franklin, Methods ofmathematical economics, Springer-Verlag, New York, 1980. [Fro] Z. Frolik, Conceming topological convergence of sets, Czech. Math. J. 10 (1960), 168-180. [FV] M. Furi and A. Vignoli, About well-posed optimization problems for functionals in metric spaces, J. Optim. Theory Appl. 5 (1970), 225-229. [Gar] A. Garkavi, The best possible net and the best possible cross section of a set in a normed space, Amer. Math. Soc. Transl. 39 (1964), 111-132. [Geo] P. Georgiev, The strong Ekeland variational principle, the strong drop theorem and applications, 1. Math. Anal. Appl. 131 (1988), 1-21. [Gll] J. Giles, Convex analysis with application in differentiation of convex functions, Research Notes in Math. # 58, Pitman, London, 1982. [Gli] I. Glicksberg, A further generalization of the Kakutani fixed point theorem, with application to Nash equilibrium points, Proc. Amer. Math. Soc. 3 (1952), 170-174. [GK] G. Godefroy and N. Kalton, The ball topology and its applications, Contemporary Math. 85 (1989), 195-237. [Gra] A. Granas, KKM-Maps and their application to nonlinear problems, in the Scottish Book, R. D. Mauldin, ed., Birkhäuser, Boston, 1981. [Gco] G. Greco, Decomposizioni di semifiltri e T-limiti sequenziali in reticoli completamente distributivi, Ann. Mat. Pura Appl. 437 (1984), 61-82. [GP] H. Greenberg and W. Pierskalla, Quasiconjugate functions and surrogate duality, Cahiers du Centre d'etudies de Rech. Oper. 15 (1973),437-448. [Gru] P. Gruber, Results of Baire category type in convexity, in Discrete geometry and convexity, Annals ofthe New York Acad. Sci. 440 (1985),163-169. [Had] G. Haddad, Monotone trajectories of differential incIusions and functional differentiapnclusions with memory, Israel J. Math. 39 (1981), 83-100. [Hahl] H. Hahn, Uber halbstetige und unstetige Funktionen, Sitzungsberichte Akad. Wiss. Wien Abt. IIa 126 (1917), 91-110. [Hah2] =---==--=--' Reelle Funktionen I, Leipzig, 1932. [Hal] B. Halpem, Fixed point theorems for set-valued maps in infinite dimensional spaces, Math. Ann. 189 (1970), 87-98. [Han] R. HanselI, First class selectors for upper semi-continuous multifunctions, J. Funct. Anal. 75 (1987), 382-395. [HJLR] R. HanselI, J. Jayne, L. Labuda, and C. Rogers, Boundaries of and selectors for upper semi-continuous set-valued functions, Math. Z. 189 (1985), 297- 318. [HJT] R. HanselI, 1. Jayne, and M. Talagrand, First class selectors for weakly upper semi-continuous multi-valued maps in Banach spaces, J. Reine Angew. Math. 361 (1985),201-220. [Hau] F. Hausdorff, Mengenlehre, W. de Gruyter, Berlin and Leipzig, 1927. BIBLIOGRAPHY 323 [HaM] A. Haworth and R. McCoy, Baire spaces, Dissertationes Math. #141, Polish Academy of Sciences, W arsaw, 1977. [Hesi] e. Hess, Contributions al'etude de la mesurabilite, de la loi de probabilite, et de la convergence des multifunctions. These d'etat, Universite Montpellier II, 1986. [Hes2] --,-__" Measurability and integrability of the weak upper limit of a sequence of multifunctions, Sem. d'Anal. Convexe Montpellier 18 (1991), expose N° 10. [Hes3] _---,-_, Epi-convergence of sequences of normal integrands and strong consistency of the maximum likelihood estimator, Cahiers de Math. de la Decision N° 9121, CEREMADE, Paris, 1992. [HU] E. Hiai and H. Umegaki, Integrals, conditional expectations and martingales of multivalued functions, J. Multivariate Anal. 7 (1977), 149-182. [Hld] W. Hildenbrand, Core and equilibria of a large economy, Princeton Univ. Press, Princeton, NJ., 1974. [Hil] L. HilI, Properties of certain aggregate functions, Amer. J. Math. 49 (1927), 419-432. [Hirn] e. Himmelberg, Measurable relations, Fund. Math. 87 (1975), 53-72. [HJV] e. Himmelberg, M. Jacobs, and F. Van Vleck, Measurable multifunctions, selectors, and Filippov's implicit function lemma, J. Math. Anal. Appl. 25 (1969), 276-284. [HPV] e. Himmelberg, T. Parthasarathy, and F. Van Vleck, On measurable relations, Fund. Math. 111 (1981), 161-167. [HV] e. Himmelberg and F. Van Vleck, Existence of solutions for generalized differential equations with unbounded right-hand side, J. Differential Equations 61 (1986),295-320. [H-U1] J.-B. Hiriart-Urruty, Lipschitz r-continuity of the approximate subdifferential of a convex function, Math. Scand. 47 (1980), 123-134. [H-U2] -=-::-:--:::-::-:-' Extensions of Lipschitz functions, J. Math. Anal. Appl. 77 (1980), 539-554. [Hol] L. Hohi, Hausdorff metric convergence of continuous functions, in Proc. sixth Prague topological symposium 1986, Z. Frolik ed., Heldermann Verlag, Berlin, 1988. [H02] __--::' The Attouch-Wets topology and a characterization of normable linear spaces, Bull. Austral. Math. Soc. 44 (1991), 11-18. [HL] L. Holli and R. Lucchetti, Equivalence among hypertopologies, preprint. [HM] L. Hohi and R. McCoy, The Fell topology on qX), in Proc. 1990 Conf. on General Topology & Appl., Long Island University, 1990, New York Acad. Sci., New York, 1992. [Holl] R. Holmes, A course in optimization and best approximation, LNM #257, Springer-Verlag, New York, 1972. [Ho12] =-=--:--' Geometric functional analysis, Springer-Verlag, New York, 1975. [How] H. Holwerda, Closed-graph and closed epigraph theorems in general topology : a unified approach, preprint. [Hö] L. Hörmander, Sur la fonction d'appui des ensembles convexes dans une espace localement convexe, Arkiv för Mat. 3 (1954), 181-186. [Huk] M. Hukuhara, Sur l'application semi-continue dont la valeur est un compact convexe, Funckcial. Ekvac. 10 (1967), 43-66. [Hur] W. Hurewicz, Über stetige Bilder von Punktmengen, Proc. Akad. Amsterdam 29 (1926), 1014-1017. [Hut] J. Hutchinson, Fractals and self-similarity, Indiana Univ. Math. J. 30 (1981), 713-747. [10] A. Ioffe, Survey of measurable selection theorems : Russian literature supplement, SIAM J. Control Optim. 16 (1978), 728-732. [Is] J. Isbell, Uniform spaces, Amer. Math. Soc., Providence, 1964. [IN] E. Ivanov and R. Nehse, Relations between generalized concepts of convexity and conjugacy, Math. Oper. für Stat, Sero Optim 13 (1982),9-18. 324 BIBLIOGRAPHY [Jac] M. Jacobs, On the approximation of integrals of multi-valued functions, SIAM J. Control Optim. 7 (1969),158-177. [IRl] J. Jayne and C. Rogers, Upper-semicontinuous set-valued functions, Acta Math. 149 (1982), 87-125. [JR2] -=-::=--_, Borel selectors for upper-semicontinuous multi-valued functions, J. Funct. Anal. 56 (1984), 279-299. [Jol] J.-L. Joly, Une familIe de topologies sur l'ensemble des fonctions convexes pour lesquelles la polarite est bicontinue, 1. Math. Pures Appl. 52 (1973), 421- 441. [KakI] S. Kakutani, A generalization of Brouwer's fixed-point, Duke Math. J. 8 (1941),457-459. [Kak2] ::--_,.--' Topological properties of the unit sphere of a Hilbert space, Proc. Imp. Acad. Tokyo 19 (1943), 269-271. [Kal] P. KalI, Approximation to optimization problems : an elementary review, Math. Oper. Res. 11 (1986),9-18. [KM] R. Kallman and R. Mauldin, A cross section theorem and an application to C* algebras, Proc. Amer. Math. Soc. 69 (1978), 57-61. [Kat] T. Kato, Perturbation theory for linear operators, Springer-Verlag, New York, 1966. [Keel] J. Keesling, Normality and properties related to compactness in hyperspaces, Proc. Amer. Math. Soc. 24 (1970), 760-766. [Kee2] =---;-:=--::-' On the equivalence of normality and compactness in hyperspaces, Pacific 1. Math. 33 (1970),657-667. [KR] K. Keimel and W. Roth, A Korovkin type approximation theorem for set• valued functions, Proc. Amer. Math. Soc. 104 (1988), 819-824. [Kenl] P. Kenderov, Monotone operators in Asplund spaces, C. R. Acad. Bulg. Sei 30 (1977), 963-964. [Ken2] ---,-_---" Most of the optimization problems have unique solution, Proc. Oberwohlfach Conf. on Parametric Optimization, B. Brosowski and F. Deutsch, eds., Inter. Series Num. Math. vol. 72, Birkhäuser-Verlag, 1984, 203-216. [KT] E. Klein and A. Thompson, Theory of correspondences, Wiley, New York, 1984. [KKM] B. Knaster, K. Kuratowski, and S. Mazurkiewicz, Ein Beweis des Fixpunktsatzes für n-dimensionale Simplexe, Fund. Math. 14 (1929), 132- 137. [Kö] G. Köthe, Topological vector spaces I, Springer-Verlag, Berlin, 1969. [Kr] H. Krüger, Aremark on the lower semicontinuity of the set-valued metric projection,1. Approx. Theory 28 (1980), 83-86. [Kul] K. Kuratowski, Sur les espaces completes, Fund. Math. 15 (1930), 301-309. [Ku2] ___, Les fonctions semi-continues dans l'espace des ensembles fermes, Fund. Math. 18 (1932), 148-159. [Ku3] K. Kuratowski, Topology, vol. 1, Academic Press, New York, 1966. [KR-N] K. Kuratowski and C. RylI-Nardzewski, A general theorem on selectors, Bull. Acad. Pol. Sci. 13 (1965), 397-402. [Lab] I. Labuda, On a Theorem of Choquet and Dolecki, J. Math. Anal. Appl. 126 (1987), 1-10. [Labl] J. Labrache, Hypertopologies recentes; quelques resultats de stabilite en analyse convexe. These, Universite de Montpellier n, 1992. [Lah2] ___, Slice topology, topologies intermediares, approximees Baire• Wijsman et Moreau-Yosida, et applications aux problemes d' optimisation convex, Sem. d'Anal. Convexe Montpellier 22 (1992), expose N° 3. [LR] J.-M. Lasry and R. Robert, Degre pour les fonctions multivoques et applications, C. R. Acad. Sei. Paris 280 (1975), 1435-1438. [Las] M. Lassonde, On the use of KKM multifunctions in fixed point theory and related topics, 1. Math. Anal. Appl. 97 (1983), 151-201. BIBLIOGRAPHY 325 [Lav] M. Lavie, Contribution a l'etude de la convergence de sommes d'ensembles aleatores independants et martingales multivoques. These, Universite Monpellier n, 1990. [Lay] S. Lay, Convex sets and their application, Wiley, New York, 1982. [Lea] S. Leader, On completion of proximity spaces by local clusters, Fund. Math. 48 (1960), 201-216. [Lec1] A. Lechicki, On bounded and subcontinuous multifunctions, Pacific 1. Math. 75 (1976), 191-197. [Lec2] ___" A generalized measure of noncompactness and Dini's theorem for multifunctions, Rev. Roumaine Math. Pures Appl. 30 (1985), 221-227. [LL] A. Lechicki and S. Levi, Wijsman convergence in the hyperspace of a metric space, Bull. Uno Mat. HaI. (7) I-B (1987), 439-452. [lZ] C. Lemarechal and 1. Zowe, The eclipsing concept to approximate a multi• valued mapping, Optimization 22 (1991), 3-37. [LLP] S. Levi, R. Lucchetti, and J. Pelant, On the infimum of the Hausdorff and Vietoris topologies, ProC. Amer. Math. SOC. 118 (1993), 971-978. [LM] P. Loridan and J. Morgan, A theoretical approximation scheme for Stackelberg problems, 1. Optim. Theory Appl. 61 (1989), 95-110. [Lub] R. G. Lubben, Concerning limiting sets in abstract spaces, Trans. Amer. Math. SOC. 30 (1928), 668-685. [LP1] R. Lucchetti and F. Patrone, Hadamard and Tyhonov well-posedness of a certain class of convex functions, 1. Math. Anal. Appl. 88 (1982), 204-215. [LP2] ___, Closure and upper semicontinuity results in mathematical programming, Nash and economic equilibria, Optimization 17 (1986), 619- 628. [LSS] R. Lucchetti, P. Shunmugaraj, and Y. Sonntag, Recent hypertopologies and continuity of the value function and of the constrained level sets, to appear, Numer. Funct. Anal. Optim. [McA] B. McAllister, Hyperspaces and multifunctions , the first half century (1900- 1950), Nieuw Arch. Wisk. 26 (1978), 309-329. [McCl] R. McCoy, Baire spaces and hyperspaces, Pacific J. Math. 58 (1975), 133- 142. [McC2] _-:-:::--" Epi-topology on spaces of lower semicontinuous functions into partially ordered spaces and hyperspaces, to appear, Set-Valued Analysis. [MN1] R. McCoy and I. Ntantu, Topological properties of spaces of continuous functions, LNM #1315, Springer-Verlag, Berlin, 1988. [MN2] -:-:-~_,' Properties of C(X) with the epi-topology, Boll. Uno Mat. Hal' 6-B (1992), 507-532. [MB] L. Mclinden and R. Bergstrom, Preservation of convergence of sets and functions in finite dimensions, Trans. Amer. Math. Soc. 268 (1981), 127-142. [MS] P. McMullen and G. Shephard, Convex polytopes and the upper bound conjecture, Cambridge Univ. Press, Cambridge, 1971. [MR] A. Maitra and B. Rao, Selection theorems and the reduction principle, Trans. Amer. Math. Soc. 202 (1975), 57-66. [Mar] M. Marjanovic, Topologies on collections of closed subsets, Publ. lost. Math. (Beograd) 20 (1966), 125-130. [MV] M. Martelli and A. Vignoli, On the differentiation of multivalued maps, Boll. Un. Mat. Hal. 10 (1974), 701-712. [Mat] G. Matheron, Random sets and integral geometry, Wiley, New York, 1975. [Mtz] M. Matzeu, Su un tipo di continuita dell' operatore subdifferenziale, Boll. Un. Mat. HaI. 14-B (1977),480-490. [Mec] E. Meccariello, Approximation of the e-subdifferential, Rend. dell' Accad. Sci. Fis. Mat. Napoli 57 (1991), 79-100. [Micl] E. Michael, Topologies on spaces of subsets, Trans. Amer. Math. Soc. 71 (1951), 152-182. 326 BIBLIOGRAPHY [Mic2] -::-::-:,.--_, Selected selection theorems, Amer. Math. Monthly 63 (1956), 233- 238. [Mic3] =-c::-:::--' Continuous selections, I. Ann. Math. 63 (1956), 361-382. [Min] G. Minty, Monotone (nonlinear) operators in a Hilbert space, Duke Math. J. 29 (1962), 341-348. [Morl] J.-J. Moreau, Inf-convolutions des fonctions numeriques sur un espace vectoriel, C. R. Acad. Sei. Paris 256 (1963), 5047-5049. [Mor2] ::-::--:--:-_' Fonctionelles convexes, lecture notes, Seminaire Equations aux Derivees Partielles, College de France, 1966. [Mosi] U. Mosco, Convergence of convex sets and of solutions of variational inequalities, Adv. in Math. 3 (1969),510-585. [Mos2] _--,---,::-::' On the continuity of the Young-Fenchel transform, 1. Math. Anal. Appl. 35 (1971), 518-535. [Mos3] :-:::-~_, Implieit variational problems and quasi variational inequalitites, in Nonlinear operators and the caleulus of variations, Bruxelles, 1975, LNM# 543, Springer-Verlag, Berlin, 1976. [Mrl] S. Mrowka, On the convergence of nets of sets, Fund. Math. 45 (1958), 237- 246. [Mr2] ___, On normal metrics, Amer. Math. Monthly 72 (1965), 998-1001. [Mr3] ___, Some comments on the space of subsets, in Set-valued mappings, selections, and topological properties of 2x, in Proc. Conf. SUNY Buffalo 1969, W. Fleischman, ed., LNM #171, Springer-Verlag, Berlin, 1970,59-63. [Nac] L. Nachman, Hyperspaces ofproximity spaces, Math. Scand. 23 (1968),201- 213. [Nad] S. Nadler, Hyperspaces of sets, Dekker, New York, 1978. [Nag] J. Nagata, On the uniform topology of bicompactifications, 1. Inst. Polytech. Osaka 1 (1950), 28-38. [Nail] S. Naimpally, Graph topology for function spaces, Trans. Amer. Math. Soc. 123 (1966), 267-272. [Nai2] -:::-----:-.,--', Topological convergence and uniform convergence, Czech. Math. 1. 37 (1987), 608-612. [Nai3] ~~-:' Hyperspaces and function spaces, Q & A in general topology 9 (1991), 33-59. [NS] S. Naimpally and P. Sharma, Fine uniformity and the locally fmite hyperspace topology on 2x, Proc. Amer. Math. Soc. 103 (1988),641-646. [NW] S. Naimpally and B. Warrack, Proximity spaces, Cambridge University Press, Cambridge, 1970. [NV] S. Nedev and V. Velov, Some properties of selectors, C. R. Acad. Sci. Bulg. 38 (1985), 1593-1596. [Neu] J. von Neumann, Zur Algebra der funktional Operationen und Theorie der normalen Operatoren, Math. Ann. 102 (1929-1930), 370-427. [Norl] T. Norberg, Convergence and existence of random set distributions, Annals ofProb. 12 (1984), 726-732. [Nor2] __-,' Random capaeities and their distributions, Probab. Th. Rel. Fields 73 (1986), 281-297 [Nor3] -:-:-:--_, On Vervaat's sup vague topology, Arkiv för Mat. 28 (1990), 139- 144. [Nur] E. Nurminski, On differentiability of multifunctions, Kybemika 5 (1978),46- 48. [OBTV] G. O'Brien, P. Torfs, and W. Vervaat, Stationary self-similar extremal processes, Probab. Th. Rel. Fields. 87 (1990), 97-119. [Olel] C. Olech, Extremal solutions of a control system, J. Differential Equations 2 (1966),74-101. [Ole2] ___, Approximation of set-valued functions by continuous functions, Colloq. Math. 19 (1968), 285-293. [Ole3] -::-----,-_, Existence theory in optimal control, in Control theory and topics in functional analysis, Vienna, 1976, Inter. Atomic Energy Agency, 1976. BIBLIOGRAPHY 327 [OIc4] ___, Existence of solutions of non convex orientor fields, Boll. Uno Mat. Ital. 11 (1975), 189-197 [Par] T. Parthasarathy, Selection theorems and their application, LNM #263, Springer-Verlag, Berlin, 1972. [Pe1] J.-P. Penot, The drop theorem, the petal theorem, and Ekeland's variational prineiple, Nonlinear Anal. 10 (1986), 813-822. [Pe2] ___, The cosrnic Hausdorff topology, the bounded Hausdorff topology, and continuity ofpolarity, Proc. Amer. Math. Soc. 113 (1991),275-286. [Pe3] ___, Topologies and convergences on the space of convex functions, Nonlinear Anal. 18 (1992), 905-916. [PT] J.-P. Penot and M. Thera, Serni-continuous mappings in general topology, Arch. Math. 38 (1982), 158-166. [PV] J.-P. Penot and M. Volle, Another duality scheme for quasiconvex problems, in Intl. Series ofNumerical Math. vol. 84, Birkhäuser-Verlag, Basel, 1988. [Per] W. Pervin, Equinormal proxirnity spaces, Indag. Math. 26 (1964),152-154. [Ph] R. Phelps, Convex functions, monotone operators, and differentiability, LNM #1364, Springer-Verlag, Berlin, 1989. [Poli] R. Poliquin, Proto-differentiation of subgradient set-valued mappings, Canad. Math. J. 42 (1990), 520-532. [Po12] ___, An extension of Attouch's theorem and its application to second• order epi-differentiation of convex composite functions, Trans. Amer. Math. Soc. 332 (1992), 861-874. [Pom] D. Pompeiu, Sur la continuite des fonctions de variables complexes, Ann. Fac. Sei. Univ. Toulouse 7 (1905), 265-315. [Popi] H. Poppe, Eine Bemerkung über Trennungsaxiome im Raum der abgeschlossenen Teilmengen eines topologischen Raumes, Arch. Math. 16 (1965), 197-199. [Pop2] ___, Einige Bemerkungen über den Raum der abgeschlossen Mengen, Fund. Math. 59 (1966), 159-169. [Pop3] ___, Über Graphentopologien für Abbildungsräume. 1., BuH. Acad. Polon. des Sei. 15 (1967),71-80. [Pry] J. D. Pryce, A device of R. J. Whitley's applied to pointwise compactness of continuous functions, ProC. London Math. Soc. 23 (1971),532-546. [PR] K. Przeslawski and L. Rybinski, Concepts of lower-semicontinuity and continuous selections for convex-valued multifunctions, 1. Approx. Theory 68 (1992), 262-282. [Rä] H. Rädstr~m, An imbedding theorem for spaces of convex sets, Proc. Amer. Math. Soc. 3 (1952), 165-169. [Rai] J. Rainwater, Spaces whose finest uniformity is metric, Pacific J. Math. 9 (1959), 567-570. [Rei] S. Reich, Approximate selections, best approximations, fixed points and invariant sets, J. Math. Anal. Appl. 62 (1978),104-113. [Rev] J. Revalski, Well-posedness of optimization problems : A survey, in Functional analysis and approximation, in ProC. Conf. Bagni di Lucca, Italy, P.-L. Papini, ed., Pitagora Editrice, Bologna, 1988. [RZ] J. Revalski and N. Zhivkov, WeH-posed optimization problems in metric spaces, to appear, 1. Optim. Theory Appl. [Ria] H. Riahi, Quelques resultats de stabilite en analyse epigraphique : approche topologique et quantitative. These, Universite de Montpellier TI, 1989. [Ric] B. Ricceri, On multifunctions with convex graph, Atti Accad. Naz. Lincei Rend. Cl. Sci. Mat. Natur. 77 (1985), 1-7. [Rbt] R. Robert, Convergence de fonctionelles convexes, 1. Math. Anal. Appl. 45 (1974), 533-555. [RV] A. W. Roberts and D. Varberg, Convex functions, Academic Press, New York, 1973. [Robl] S. Robinson, Normed convex processes, Trans. Amer. Math. Soc. 174 (1974), 127-140. 328 BIBLIOGRAPHY [Rob2] -::-_=-" Regularity and stability for eonvex multivalued funetions, Math. Oper. Res. 1 (1976), 130-143. [Roc1] R. T. Roekafellar, Monotone proeesses of eonvex and eoneave type, Memoirs of the Amer. Math. Soe. #77, Providenee, Rhode Island, 1967. [Roe2] -=-::--::-..,.-" Measurable dependenee of eonvex sets and funetions on parameters, J. Math. Anal. Appl. 28 (1969),4-25. [Roc3] :::--:-:::---::-' On the maximal monotonicity of the subdifferential mappings, Pacific 1. Math. 33 (1970),209-216. [Roc4] :::-_---,., Convex analysis, Prineeton University Press, Prineeton, New Jersey, 1970. [Roe5] ::-:--:-:_" Integral funetionals, normal integrands and measurable seleetions, in Nonlinear operators and the ealculus of variations, LNM #543, Springer• Verlag, Berlin, 1976, 157-207. [Roc6] _-:---:--" Proto-differentiability of set-valued mappings and its applieation in optimization, in Analyse non lineaire, H. Attoueh, 1.-P. Aubin, F. Clarke, and I. Ekeland, eds., Gauthier-Villars and C. R. M. Universite de Montreal, 1989, 449-482. [RW] R. T. Roekafellar and R. Wets, Variational systems, an introduetion, in Multifunetions and integrands, in Proe. Conf. Catania 1983, G. Salinetti, ed., LNM #1091, Springer-Verlag, Berlin, 1984, 1-54. [SB] M. F. Saint Beuve, On the extension ofvon Neumann-Aumann's Theorem, J. Funet. Anal. 17 (1974), 112-129. [SWl] G. Salinetti and R. Wets, On the relationship between two types of eonvergenee for eonvex funetions, J. Math. Anal. Appl. 60 (1977), 211-226. [SW2] --:7""-~' On the eonvergenee of sequenees of eonvex sets in finite dimensions, SIAM Rev. 21 (1979), 18-33. [SW3] ___, On the eonvergenee of closed-valued measurable multifunetions, Trans. Amer. Math. 266 (1981), 275-289. [SW4] _-:-_' Convergenee in distribution of measurable multifunetions, (random sets), normal integrands, stoehastic proeesses and stoehastie infima, Math. Oper. Res. 11 (1986),385-419. [Sen] BI. Sendov, Hausdorff approximations, Kluwer, Dordreeht, Holland, 1990. [Sh] P. Shunmugaraj, On stability aspeets in optimization. Thesis, Indian Institute of Teehnology, Bombay, 1990. [ShP] P. Shunmugaraj and D. Pai, On stability of approximate solutions of minimization problems, to appear, Numer. Funet. Anal. Optim. [Sie] W. Sierpinski, Sur l'inversion du theoreme de Bolzano-Weierstrass generalise, Fund. Math. 34 (1947), 155-156. [Simi] S. Simons, Two-funetion minimax theorems and variational inequalities for funetions on eompaet and noneompaet sets, with some eomments on fixed• point theorems, Proc. Symp. Pure Math. 45 (1986), 377-392. [Sim2] --0'"___ ...,....-', On a fixed-point theorem of Cellina, Atti Aeead. Naz. Lineei Rend. Cl. Sei. Mat. Natur. 80 (1986),8-10. [Sim3] ~=-_" Subtangents with eontrolled slope, to appear, Nonlinear Anal. [Sini] I. Singer, Some remarks on approximative eompaetness, Rev. Roumaine Math. Pures Appl. 9 (1964),167-177. [Sin2] ___, On Set-valued metrie projeetions, in Linear Operators and approximation, P. Butzer, J. Kahane, and B. Sz.-Nagy, eds., Inter. Series Num. Math. vol. 20, Birkhäuser-Verlag, Basel, 1972,217-233. [Sin3] -::-:-::--:::::-:-' The theory of best approximations and funetional analysis, CBMS #13, SIAM, Philadelphia, 1974. [Sin4] _--::-_, Conjugate operators, in Seleeted topies in operations researeh and mathematieal eeonomies, G. Hammer and D. Pallasehke, eds., Leeture Notes in Eeon. and Math. Systems #226, Springer-Verlag, Berlin, 1984,80-97. [Sio] M. Sion, On general minimax theorems, Pacifie. 1. Math. 8 (1958),171-176. [Sml] R. Smithson, Some general properties of multivalued funetions, Pacifie. J. Math. 15 (1965), 681-703. BIBLIOGRAPHY 329 [Sm2] ___, Multifunctions, Nieuw Arch. Wisk. 20 (1972), 31-53. [Sm3] ~-::--_, Subcontinuity for multifunctions, Pacific J. Math. 61 (1975),283- 288. [Sm4] -=-=-=-=-::-::-' First countable hyperspaces, Proc. Amer. Math. Soc. 56 (1976), 325-328. [So] Y. Sonntag, Convergence au sens de U. Mosco : theorie et applications a l'approximation des solutions d'inequations. These, Universite de Provence, Marseille, 1982. [SZl] Y. Sonntag and e. Zalinescu, Scalar convergence of convex sets, 1. Math. Anal. Appl. 164 (1992), 219-241. [SZ2] _~.,.--" Set convergences: An attempt of classification, in Proc. Intl. Conf. on Diff. Equations and Control Theory, Iasi, Romania, August, 1990. Revised version, to appear, Trans. Amer. Math. Soc. [SZ3] ::-::--:----,---,-' Set convergences : a survey and a classification, to appear, Set• Valued Anal. [Soul] M. Soueycatt, Epi-convergence et convergence des sections. Application ala stabilite des e-points-selles de fonctions convexes-concaves. Publications AVAMAC 2 (1987) 7.01-7.40. [Sou2] ~-:----:-' Analyse epilhypo-graphique des problems de points-seIles. These, Universite de Montpellier 11, 1991. [Spe] W. Spears, A generalization ofthe Stone-Weierstrass Theorem to multi-valued functions, J. Nat. Sci. and Math. (Labore) 9 (1969), 201-208. [Ste1] e. Stegall, A class of topological spaces and differentiation of functions in Banach spaces, Proc. Conf. on Vector Measures and Integral Representations of Operators, and on Functional Analysis (Banach Space Geometry), Essen, 1982, Vorlesungen Fachbereich Math. Univ. Essen #10 (1983), 63-67. [Ste2] __--,-, Gateaux differentiation of functions on a certain class of Banach spaces, in Functional Analysis : Surveys and recent results m, K. Bierstadt and B. Fuchssteiner, eds., Elsevier North-Holland, Amsterdam, 1984. [Str] W. Strother, Fixed points, fixed sets, and M -retracts, Duke Math. J. 22 (1955),551-556. [Su] F. Sullivan, A characterization of complete metric spaces, Proc. Amer. Math. Soc. 83 (1981), 345-346. [Toa] Gh. Toader, On a problem of Nagata, Mathematica (Cluj) 20 (1978), 78-79. [Tom] V. Toma, Quelques problemes de mesurabilite des multifonctions, Sem. d'Anal. Convexe Montpellier 13 (1983), expose N° 6. [Ton] H. Tong, Some characterizations of normal and perfectly normal spaces, Duke Math. J. 19 (1952), 289-292. [Ts] M. Tsukada, Convergence of best approximations in a smooth Banach space, J. Approx. Theory 40 (1984), 301-309. [Ty] A. Tyhonov, On the stability of the functional optimization problem, USSR Comp. Math. and Math. Phys. 6 (1966), 28-33. [Ur] e. Ursescu, Multifunctions with closed convex graph, Czech. Math. J. 25 (1975),438-441. [Vai] I. Vainstein, On closed mappings of metric spaces, Dokl. Acad. Nauk Soviet Union 57 (1947),419-421. [Vld1] M. Valadier, Contribution al'analyse convexe. These, Paris, 1970. [Vld2] ___ -:-~" Convex integrands on Souslin locally convex spaces, Pacific J. Math 59 (1975), 267-276, [Val] F. Valentine, Convex Sets; McGraw-Hill, 1964. [VC] B. Van Cutsem, Martingales de convexes fermes aleatoires en dimension fmie, Ann. Inst. H. Poincare Sec. B 8 (1972), 365-385. [Vau] H. Vaughan, On locally compact metrizable spaces, Bull. Amer. Math. Soc. 43 (1937), 532-535. [VV1] A. Verona and M. E. Verona, Locally efficient monotone operators, Proc. Amer. Math. Soc. 109 (1990), 195-204. 330 BIBLIOGRAPHY [W2] ___, Epi-topologies and e-subgradients, preprint. [Verl] W. Vervaat, Une compactification des espaces fonctionnels C et D; une alternative pour la demonstration des theoremes limites fonctionnels, C. R. Acad. Sei. Paris 292 (1981), 441-444. [Ver2] ___, Random upper semicontinuous functions and extremal processes. Report # MS-8801, Center for Math. and Comp. Sei., Amsterdam, 1988. [Viel] L. Vietoris, Stetige Mengen, Monatsh. für Math. und Phys. 31 (1921), 173- 204. [Vie2] -;:-:=:--:::" Bereiche zweiter Ordnung, Monatsh. für Math. und Phys. 32 (1922), 258-280. [Vitl] R. Vitale, Approximation of convex set-valued functions, 1. Approx. Theory 26 (1979), 301-316. [Vit2] -=-=-:::--:=-==' 1.P metrics for compact convex sets, 1. Approx. Theory 45 (1985), 280-287. [VI] L. Vlasov, Approximative properties of sets in normed linear spaces, Russian Math. Surveys 28 (1973), 1-66. [Voll] M. Volle, Convergence en niveaux et en epigraphs, C. R. Acad. Sei. Paris 299 (1984),295-298. [VoI2] ::-7:--" Conjugaison par tranches, Ann. Mat. Pura Appl. 139 (1985), 279- 311. [VoI3] --:::-:---0-.' Contributions a la dualite en optimisation et a l'epi-convergence. These d'etat, Universite de Pau, 1986. [Wag] D. Wagner, Survey of measurable selections, SIAM J. Control Optim. 15 (1977), 859-903. [WW] D. Walkup and R. Wets, Continuity of some convex-cone valued mappings, Proc. Amer. Math. Soc. 18 (1967), 229-235. [War] A. Ward, A counter-example in uniformity theory, Proc. Camb. Phil. Soc. 62 (1966), 207-208. [Wat] W. Waterhouse, On UC spaces, Amer. Math. Monthly 72 (1965), 634-635. [Wei] W. Weil, An application of the centrallimit theorem for Banach-space-valued random variables to the theory ofrandom sets, Z. Wahrsch.Theorie 60 (1982), 203-208. [Wetl] R. Wets, Convergence of convex functions, variational inequalities, and convex optimization problems, in Variational inequalities and complementarity problems, R. Cottle, F. Gianessi, and 1.-L. Lions, eds., Wiley, Chichester, 1980, 375-403. [Wet2] ".-....,-_, A formula for the level sets of epi-limits and some applications, in Mathematical theories of optimization, 1. Cecconi and T. Zolezzi, eds., LNM #979, Springer-Verlag, Berlin, 1983,256-268. [Wic] K. Wicks, Fraetals and hyperspaces, LNM # 1492, Springer-Verlag, Berlin, 1991. [Wij] R. Wijsman, Convergence of sequences of eonvex sets, eones, and funetions, II, Trans. Amer. Math. Soc. 123 (1966), 32-45. [Wil] S. Willard, General topology, Addison-Wesley, Reading, Mass. 1970. [Zab] S. Zabell, Moseo eonvergenee in loeally eonvex spaees, J. Funet. Anal. 110 (1992), 226-246. [Zar] C. Zarankiewiez, Sur les points de division dans les ensembles connexes, Fund. Math. 9 (1927), 124-171. [Zol] T. Zolezzi, Stability analysis in optimization, in Optimization and related fields, Proe. Conf. Erice 1984, R. Conti, E. De Giorgi, and F. Gianessi, eds., LNM # 1190, Springer-Verlag, Berlin, 1986,397-419. [Zorl] L. Zoretti, Sur les fonetions analytiques uniformes qui possedent un ensemble diseontinu parfait de points singuliers, J. Math. Pures et Appl. 1(1905), 1-51. [Zor2] -::-=-...,...,.,::-::-::" Un theoreme de la theorie des ensembles, BuH. Soc. Math. Franee 37 (1909), 116-119. Symbols and Notation 1. Sets and functions in the abstract z+ the set of positive integers 1 R the set of real numbers 1 AC the complement of the set A AUB the union of sets A and B 1 AnB the intersection of sets A and B A\B the set difference of A and B 1 AxB the product of sets A and B 1 IliE I Xi the product of a family of sets {Xi: i E I} 8 idA the identity function for a set A 1 iA the inclusion map for a set A 8 BA the set of all functions from A to B rn/(A) the infimal value of an extended real function j on the set A 110 Grj the graph of a function j 56 epij the epigraph of an extended real function j 13 hypoj the hypo graph of an extended real function j 13 A ie/Ii the meet of a family of functions lfi: i E I} 14 V ie/Ii the join of a family of functions {fi: i E I} 14 slv (j;a) the sublevel set of a function j at height a: {x :f(x)::;; a} 14 domj the effective domain of a proper function j 15 jlA the restrietion of a function j to A 16 I(· ,A) the indicator function of a set A 16 Argminj the set of minimizers of j 15 iV the sup-integral operator 20 dV the sup-derivative operator 20 T:X:4 Y T is a multifunction from X to Y 184 domT the domain of T: {x EX: T(x) -:I- 0} 184 mgT the range of T: {y E Y: 3 x E X with Y E T(x)} 184 GrT the graph of T: {(x,y) E X x Y: Y E T(x)} 184 T the single valued function associated with the multifunction T 195 T(A) the image of A<= X under T: UaeA T(a) 184 T-l(B) the preimage of B<= Y under T: {x EX: T(x) n B -:I- 0} 184 2. Sets and functions in metric or topological spaces cl A closure of the set A 1 int A interior of the set A 1 bd A boundary of the set A A I the set of accumulation points of A 331 332 SYMBOLS AND NOTATION n(x) the open neighborhoods of a point x 2 't'ucb the topology of uniform convergence on bounded sets 79 't'co the compact-open topology l3 duc the metric of uniform convergence 3 ducb the metric of uniform convergence on bounded sets 79 SYMBOLS AND NOTA nON 333 Frac T the active boundary of a multifunction T 188 B(X) the Borel field of a topological space X 217 Cl®B product sigma algebra 221 E(I> the Effros sigma algebra on a family of c10sed sets L 222 3. Sets and functions in linear spaces A+B the vector sum of sets A and B in a linear space 3 AEBB the c10sure of A + B 4 A-B the vector difference of sets A and B in a linear space 3 aA the scalar multiple of a set A by a 3 coA the convex hull of a set A 21 c1coA the c10sure of the convex hull of A 21 Jl p the space of p-summable sequences 3 X* the continuous linear functionals on a normed linear space X 4 X** the continuous linear functionals on X* 8 (A()* the origin of X\X* 3-4 U\U* the c10sed unit ball of X\X* 3-4 haus,u(A,B) max {ed(A n IlU, B), ed(B n IlU, A)} 83 4 . Hyperspace topologies and set convergences 'rWd the Wijsman topology determined by the metric d 34 'rWd the lower Wijsman topology determined by the metric d 114 'r~d the upper Wijsman topology determined by the metric d 114 'rBd the ball proximal topology determined by the metric d 45 'rv the Vietoris topology 47 'r8d the proximal topology determined by the metric d 50 'rMd the bounded proximal topology determined by the metric dill C18d the dual proximal topology determined by the metric d 115 C1Md the dual bounded proximal topology determined by the metric d 115 'l(L1) the hit-and-miss topology determined by .1 52, 128 a(L1) the proximal hit-and-miss topology determined by .1 52, 128 I(L1) the family of all finite unions of members of .1 128 'rs the slice topology on the cIosed convex sets 61 'rS the dual slice topology on the weak* cIosed convex sets 64 'rAWd the Attouch-Wets topology determined by the metric d 79 'rA Wd the lower Attouch-Wets topology determined by the metric d 114 'rAWd the upper Attouch-Wets topology determined by the metric d 114 'rHd the Hausdorff metric topology determined by the metric d 85 'rHd the lower Hausdorff metric topology determined by the metric d 114 'r"lId the upper Hausdorff metric topology determined by the metric d 114 'rlf the locally finite topology 97 'rsc the scalar topology of pointwise convergence of support functionals 121 'rL the linear topology on the cIosed convex sets 122 m++ m++ .:1\ Li,R' .:1\ Li,1 cIasses of uniformly continuous functions whose sublevel sets are interposed by elements of the family 1:\.1) 129 cIasses of continuous functions whose sublevel sets are interposed by elements of the family 1:\.1) 133 'rF the Fell topology 138 LiA;. the lower cIosed limit of a net of sets 145 LsA;. the upper cIosed limit of a net of sets 145 K-limA;. the Kuratowski-Painleve limit of a net of sets 146 'rM the Mosco topology 171 M-limAn the Mosco limit of a sequence of sets 171 '1Hd the finite Hausdorff topology 225 'rA the affine topology 286 Subject Index active boundary 188 Bishop-Phelps Theorem 299 affine combination 7 Borel field 217, 223 affine function 6, 25 Borwein-V anderwerff Theorem 65 affine topology 286 bounded proximal topology 111 Alexander Subbase Theorem 53 dual 115 Alexandroff s Theorem 69 bounded Vietoris topology 120 angelic space 207 boundedly compact space 35 approximatively compact 197,268 box norm 3 Argrnin 15, 160, 170, 177, 186, 191, Brouwer's Theorem 209 265,268-269 Caratheodory function 222, 228, 234 Arzela-Ascoli Theorem 70 carrier 310 Ascoli formula 5 Castaing representation 232 Asplund space 62 category 10 Attouch's Theorem 297,314 Cech-complete space 10 Attouch-Wets convergence 79, 82-83 Cellina's Theorem 234 of cones 91, 245-248 chainable set 90 versus Hausdorff metric convergence Chebyshev 86, 90 center 207 of l.s.c. functions 235 radius 207 versus slice convergence 81, 84 set 186 and strong convergence in X* 102 Choquet-Dolecki Theorem 201 versus Wijsman convergence 80 closed sum4 Attouch-Wets topology 79 cluster point of a net 2, 145 for l.s.c. functions 235 coercivity 249 lower 114 cofinal set of indices 2 metric for 79 completed graph 197 for different metrics 93 complete metrizability uniforrnities for 81 Attouch-W ets topology 80 upper 114 Hausdorff metric topology 87 as a weak topology 117-119 slice topology 75 Aze-Penot inequality 84 Wijsman topology 71 Baire space 10 cone 21 balanced set 64 maximal point 289, 299 ball 2 normal 27, 216, 305 ball proximal topology 45 pointed 298 and Wijsman topology 45 recession 26 Banach-Alaoglu Theorem 8 conjugate function 241 Banach space 3 constraint qualification 256, 267 Bartle-Graves Theorem 230 continuity ofpolarity 61,248,279-280, Berge Maximum Theorem 202 286-287 bipolar theorem 22 continuity of restriction 261 335 336 SUBJECT INDEX continuity of set addition 113, 123-125, strict 254 152,154-155,177,263 epigraphical multifunction 199,227,255, continuum hypothesis 150 276 convex 4 epigraphical sealar multiplication 255 combination 21 epi-hypo convergence 312 function 24 epi-sum250 huH21 equicontinuous family 70 kernell12, 191 equi-Lipschitzian family 70 process 192 equi-Iower semicontinuous family 168 set 4,20 excess 28 Corradini' s Theorem 68 excess functional 106 correspondence 310 Fa set 19 Costantini-Levi-Zieminska Theorem 39 Ky Fan Inequality 215 Curtis-Schiori Theorem 308 far44 d-inc1usion 38 FeH topology 138 strict 38 compactness of 139, 149 Deutsch and Kenderov Theorem 228 convergence in 141, 146 diameter 3 extended 139 Dieudonne Interposition Theorem 233 for l.s.c. functions 164-170 Dini' s Theorem 19, 198 metrizability of 140 directed set 2 uniformity for 144 discrete family of sets 98 versus Wijsman 142 discrete set Fenchel' s equality 288 Fenchel transform 243 E- 36 uniformly 36 and Attouch-Wets convergence 247 and slice convergence 277, 279 distance functional 25 and Wijsman convergence 287 domain finite Hausdorff topology 225 effective 15 Fisher convergence 54, 307 of a multifunction 184 Drop Theorem 313 flat 4 duality mapping 187, 191,206,299 Frac 188, 190,201, 206-207 function Eberlein-Smulian Theorem 173 affme6 efficiency 311 Effros sigma algebra 222 Borel217 Efremovic Lemma 92 convex 24 Efremovic' s Theorem 100 distance 25 indicator 16 Ekeland Principle 290 analytic formulation 298 measurable 217 and Attouch-Wets convergence 300 positively homogeneous 27 and slice convergence 303 proper 15 enlargement 2 semicontinuous 13 stable under 108, 110, 113 subadditive 27 entourage9 Furi-Vignoli Theorem 268 epi-convergence 156 Gö set 10 epigraph 13 r- convergence 310 g.w.p.264 SUBJECT INDEX 337 gap 28 James' Theorem 67 gap functional88, 106-109, 121-122, join 14 175 Joly Topology 276, 313 graph topology 57 k-space 310 Hahn-Banach Theorem 27 KKM map 208 half-space 21 KKM Principle 210 Hausdorff distance 7, 85 Kakutani's Theorem 214,234 Hausdorff metric topology 85, 308 Kato gap 309 and epi-convergence 161 kernel lower 114 convex 112, 191 for different metrics 92 of a multifunction 200 versus Fell topology 143 Krein' s Theorem 173 suprema of 98 Krein-Smulian Theorem 68 and uniform convergence 85, 91 Krüger's Theorem 233 upper 114 Kuratowski-Painleve convergence 146 versus Vietoris topology 90 compactness of 149 as a weak topology 87,117-118 of epigraphs 15 versus Wijsman topology 87 versus Fell 147 hausJl 83 of multifunctions 221 Hess' Theorem 224 sequential 148, 310 hit 43 of subdifferentials 297 hit-and-miss topology 44, 52-53, 129 and weak* convergence 153 proximal 45, 52-53, 108-110, 129 versus Wijsman convergence 154 Hohi-Lucchetti Theorem 45 Kuratowski' s Theorem 7, 59 Hörmander' s Theorem 89 Kuratowski and Ryll-Nardzewski hyperplane 4 Selection Theorem 311 separating 21 lbc topology 309 supporting 21 .Jlp space 3 hyperspace 1 Leaders' s Theorem 100 admissible 1 Lebesgue number 54 extended 136 limit point of a net 2, 145 hit-and-miss 44, 129 linear topology 122, 287 proximal hit-and-miss 45, 129 Lipschitz regularization 251, 256 uniformizable 128 and Attouch-Wets convergence 261 hypograph 13 and Mosco convergence 286 indicator function 16 and slice convergence 283, 286 inf-bounded 113 local family 135 infimal convergence 310 locally convex space 8, 208 infimal convolution 250 locally finite family 95 infimal value 16 locally finite topology 87,308 infimal value functional 110, bounded 99 128-137, 145 lower closed limit 145 inf-vague topology 164 of epigraphs 158 intermediate topology 121 lowerenvelope 14, 19, 197 inward map 215 lower semicontinuous 338 SUBJECT INDEX function 13 graph of 184 multifunction 192 Hausdorff upper semicontinuous 311 Mackey topology 176 image of a set 184 marginal function 202 inverse image of a set 184 measurable graph 222, 311 Kenderov continuous 311 measurable set 216 locally bounded 200 measurable space 216 lower semicontinuous 192 measurable multifunction 218 outer semicontinuous 188 measure of noncompactness 7, 60, 91, product 185 155,268 range of 184 meet 14 subcontinuous 206 metric 2 upper hemicontinuous 311 almost convex 108 upper semicontinuous 193 Attouch-Wets 79 values of 184 compatible 3 net 1 equivalent 3 cluster point of 2, 145 Hausdorff 85 convergent 2 infinite-valued 3 limit inferior 2 with nice closed balls 142 limit point of 2, 145 product or box 3 limit superior 2 space 2 subnet 149 of uniform convergence 3 nice closed balls 142 uniformly equivalent 44 norm 3 metric projection 186, 188, 191, 197, dual 4 200-201,207,227,233,268 equivalent 3 Michael's Selection Theorem 229 Kadec 60 minimax theorem 213 operator 4 minimizing sequence 19,264 strictly convex 191 miss 44 weak* Kadec 60 strongly 44 weak*-Mackey Kadec 179 monotone convergence theorem 169 normal integrand 255 monotone operator 298 normed linear space 3 Moreau's conjugate formula 253 one-point compactification 144 Moreau-Yosida regularization 251 oscillation 20 Mosco convergence 171 outward map 215 and Mackey convergence 175 paracompact space 95 versus slice 173-174 partition of unity 96 versus Wijsman 178-182 perfectly normal space 218 Mosco topology 171 polar 22 Mrowka's Theorem 149 Polish space 69 multifunction 184 polytope 21 composition 185 proper function 15 domain of 184 proximal normal 191 efficient 311 proximinal set 186, 190 fixed point 209 proximal topology 50 SUBJECT INDEX 339 dual 115 and strong convergence in X* 102 and excess functionals 111 versus Wijsman convergence 62, 65- and gap functionals 109 68,207 and uniform convergence 57 slice topology 61 and Wijsman topologies 50 for convex functions 76, 270- pseudo-Cauchy sequence 59 272, 276-278, 295 quasi-concave function 213 dual 64, 68 quasi-convex function 113 and gap functionals 109 Radsträm Cancellation Principle 257 versus linear topology 126 recession, direction of 26 metrizability of 75 refinement 95 and Wijsman topologies 62 barycentric 233 smoothing kernel 251 reflexive Banach space 9 stability of interior 257,263 regularization 250-254, 263 starshaped 112 Lipschitz 251, 256 strong containment 43 Moreau-Yosida 251 strong minimum 264,301 relative topology 8 strongly exposing functional 269 renorming 3 subdifferential 287 and Wijsman convergence 62-68 subgradient 287 residual set of indices 2 e- 288 restriction 16 sublevel set 14 Riesz Representation Theorem 182 convergence of 155, 163, 167- saddle point 216 170, 178,239,241,273,276 scalar convergence 121 submultifunction 185 scalar topology 121 subtangent 293 Schauder Theorem 211 sum theorem 238, 259, 275 Schur space 172 sup-derivative 20 selection 228 sup-integral 20 e-approximate 228 sup-measure 20 convergence of 234 support 21 separation 21 functional 25, 66, 89, 91, 121,206- strong 21 207,227,242,287 e- 31 hyperplane 21 Separation Theorem 22 point 27 shrinking of a cover 96 Tietze Extension Theorem 58 sigma algebra 216 topologization 147 product 221 topology sigma compact 219 compact-open 13,70 slice 62 hyperspace 1 slice convergence 62, 66, 67, 109,305 infimum 9 versus Attouch-Wets convergence 81, pointwise convergence 34 84 product 8 of convex functions 275, 281, 283, relative 8 297,299 supremum 9 and normal cones 305 uniform convergence 56 340 SUBJECT INDEX uniform convergence on locally finite 120 bounded sets 79 proximal 50, 109, 111, 136 totally bounded set 7 scalar 121 translation invariance 101, 111 slice 62, 109,276,286,295 UC space 54 Vietoris 48, 113, 136 UC metrizability 59 Wijsman 34 Uniform Boundedness Principle 11 weak* topology 8 uniform continuity 128 weighted average principle 209 uniformity 9 well-posedness 264,301 base 9 Wijsman convergence 35 subbase 9 and linear functionals 41 uniformly Urysohn family 129 of multifunctions 221 unit ball 3 Wijsman topology 34 upper closed limit 145 for convex functions 75-76 of epigraphs 158 extended 145 upper envelope 20, 197 lower 114 uppersemicontinuous metrizability of 36 function 13 metric for 37 multifunction 192 for different metrics 39 Urysohn family 132 as a hit-and-miss topology 45,53 uniformly 129 suprema of 48, 50, 62-65, 68 Urysohn Metrization Theorem 3 upper 114 usco map 194 minimal 198 Vainstein Lemma 206 Vervaat's Theorem 166,178 Vietoris topology 47 basis for 53 versus Fell topology 144 versus Hausdorff metric topology 90 and Wijsman topologies 48 visibility multifunction 187, 196, 227 w.p.264 Walkup-Wets Isometry Theorem 246 weak topology 8 weak hyperspace topologies 107,223 affine 287 Attouch-Wets 117, 119 bounded dual proximal 115 bounded proximal 111,118 bounded Vietoris 120 dual proximal 115 Fell 142-143, 145 Hausdorff metric 88, 117, 118 linear 122, 127