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On Mosco Convergence of Convex Sets BULL. AUSTRAL. MATH. SOC. 52AO7, 54B20, 41A50 VOL. 38 (1988) [239-253] ON MOSCO CONVERGENCE OF CONVEX SETS GERALD BEER We present a natural topology compatible with the Mosco convergence of sequences of closed convex sets in a reflexive space, and characterise the topology in terms of the continuity of the distance between convex sets and fixed weakly compact ones. When the space is separable, the topology is Polish. As an application, we show that in this context, most closed convex sets are almost Chebyshev, a result that fails for the stronger Hausdorff metric topology. 1. INTRODUCTION About twenty years ago Mosco [28] introduced a fundamental notation of con- vergence for sequences of closed convex sets in a reflexive space, now called Mosco convergence, widely applicable to convex optimisation, the solution of variational in- equalities, and to the theory of optimal control. Specifically, a sequence {Cn) of closed convex sets in a reflexive space X is declared Mosco convergent to a closed convex set C provided: (i) at each x in C there exists a sequence (xn) convergent strongly to x such that for each n, xn 6 C'n , and (ii) whenever (n(k)) is an increasing sequence of positive integers and xn^j £ r Cn(k) f° each k, then the weak convergence of (xn(/t)) to x implies x G C. The most celebrated result on Mosco convergence is concerned with the convergence of sequences of convex functions (as identified with their epigraphs) on a reflexive space. A sequence of lower semicontinuous proper convex functions (/„) on X is declared Mosco convergent to a convex function / provided (epi /n) converges in the above sense to epi /. Locally, this means at each x in X: (i) there exists a sequence {xn) convergent strongly to x for which lim/n(:cn) = f(x), and (ii) whenever (xn) converges weakly to x, then liminf/n(zn) > f(x) [28, Lemma 1.10] Received 18 November 1987 Copyright Clearance Centre, Inc. Serial-fee code: 0004-9729/88 $A2.00+0.00. 239 Downloaded from https://www.cambridge.org/core. IP address: 170.106.202.226, on 27 Sep 2021 at 20:19:02, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0004972700027519 240 G. Beer [2] With this notion of convergence, the Young-Fenchel transform, that is, the conjugate operator, is "continuous": if (/„) is Mosco convergent to /, then {f£) is Mosco con- vergent to /* ([22, 20]). More recently, Mosco convergence has reared its head at the foundations of func- tional analysis [7]. If X is reflexive, then the norm convergence of a sequence (yn) in X* to a nonzero limit y £ X* is equivalent to the Mosco convergence of the sequence of level sets ({x S X \ (yn, x) = a}) to {x £ X: (y, x) = a}, for each real a. This is somewhat surprising, for Mosco convergence may be described without reference to norms or distance at all. In his monograph on variational convergence [2], At touch displayed a topology compatible with Mosco convergence of convex functions, and showed that the topology is Polish (second countable and completely metrisable) when the reflexive space is, in addition, separable. An induced topology on convex sets is then obtained, identifying a set with its indicator function (Mosco convergence of a sequence of closed convex sets is equivalent to the Mosco convergence of their indicator functions). The approach of Attouch is highly indirect, resting on the equivalence between Mosco convergence of sequences of functions and the pointwise convergence of their Moreau-Yosida approxi- mates, under a suitable renorming of the space. The purpose of this article is provide a direct and much simpler description of the topology of Mosco convergence for convex sets, that seems more tractible for handling questions of a geometrical nature. As a particular application, we show that with respect to this topology, most convex sets in a separable reflexive space are almost Chebyshev, a result that fails when the stronger HausdorfF metric topology is used instead. At this time, the most complete reference on Mosco convergence of setsis the thesis of Sonntag [32]. The reader may also consult [2, 30, 34] and of course, the papers of Mosco himself. 2. NOTATION AND TERMINOLOGY In the sequel, all spaces X are real normed linear spaces. We distinguish the following classes of subsets of X ; C(X) = the closed convex nonempty subsets of X K(X) = the weakly compact nonempty subsets of X O(X) — the strongly open nonempty subsets of X CL{X) = the strongly closed nonempty subsets of X We denote the closed unit ball of X by U and the origin of X by 0. If {x, xi, X2, • • •} C X and (xn) converges strongly (respectively weakly) to x , we write (respectively x = w — limxn). If A £ CL(X) and if B € CL(X), we set Downloaded from https://www.cambridge.org/core. IP address: 170.106.202.226, on 27 Sep 2021 at 20:19:02, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0004972700027519 [3] On Mosco convergence 241 d(A, B) = inf{||a- 6|| | a G A, b G B}, forgiving the abuse d(x, A) for d({x}, A). If A' is reflexive and x G X and C G C(X), then x has at least one nearest point in C. The set of nearest points to x in C is called the metric projection of x onto C, and will be denoted by P(x, C). A set C is called Chebyshev [20] (respectively almost Chebyshev [19]) if {x G X | P(K, C) is a singleton} is all of X (respectively is a dense and Gg subset of X). If X is strictly convex, then each element of C(X) is Chebyshev. For background, it is perhaps worthwhile to list the basic topologies on C(X). All of them are considered (in a more general setting) either explicitly or implicitly in a recent unifying paper of Francaviglia, Levi, and Lechicki [18]. If A G C(X) and B G C(X), then the Hausdorff distance H between the two sets is given by H{A, B) = inf{a | A + ctU D B and B + all D A}. Hausdorff distance so defined yields an infinite valued complete metric on C(X) ([9, 24]). We denote the associated topology by TH . It is well-known that Hausdorff metric convergence of a net of sets is equivalent to the uniform convergence of their distance functions; more precisely, H(A,B) = \\d(.,A)-d(.,B)\\00. A therefore weaker notion of convergence for net of sets is pointwise convergence of their distance functions. It is known that this convergence is compatible with a completely regular topology T\y on C(X), called the Wijsman topology, which is metrisable when X is separable ([18, 26]). We now turn to the standard "hit and miss" topologies in the literature. For each nonempty subset E of X , we define collections of closed convex sets E+ and E~ by the formulas E+ = {C G C(X) | C C E} and E~ = {C G C(X) | C n E ± 0}. The topologist's favourite topology of this genre is the Unite or Vietoris topology ([24, 27]) Ty , which includes as a subbase all sets of the form V~ and V+ where V G O(X) . Thus, a basic open neighbourhood of a convex set C consists of all convex sets that hit each member of a prescribed finite list of open sets, and which miss (fail to hit) a prescribed closed set. For the purposes of analysis, this topology is extremely patholog- ical. The Fell topology rp [16], also called the the topology of closed convergence [24] or the topology of set convergence [2], has as a subbase all sets of the form V~ where V G O(X) and (Kc) where K is (strongly) compact. The Fell topology seems to be well-behaved only when X is finite dimensional; in this case it is Polish (this follows Downloaded from https://www.cambridge.org/core. IP address: 170.106.202.226, on 27 Sep 2021 at 20:19:02, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0004972700027519 242 G. Beer [4] from [24, Theorem 4.5.5] and the fact that a Gs subset of a completely metrisable space is completely metrisable; see more directly the comments preceding Lemma 4.1 of [6]). By the hall topology TB [4], we mean the topology with a subbase consisting of all sets of the form V~ where V G 0(X) and (Bc)+ where B is closed ball. We note that O(X) may be replaced by the collection of open balls in the description of a subbase for Tg, justifying the terminology. It is well known that rp = TW = T~B when X is finite dimensional (see, for example, Theorem 2.3 of [4]). Finally, we may consider convergence of convex sets to mean pointwise convergence of their support functions ([12, 31]). In finite dimensions, this is neither stronger nor weaker than convergence with respect to the Fell (— Wijsman = ball) topology (see, for example, [31, p.29]. 3. THE MOSCO TOPOLOGY DEFINITION: Let X be a Banach space. The Mosco topology TM on C(X) is the topology generated by all sets of the form V~ whereV £ O(X) and (A'c)+ with K e K{X).
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