Annali di Matematica pura ed applicata (IV), Vol. CLX (1991), pp. 303-320
The Topology of the ,z-Hausdorff Distance (*).
HEDY ATTOUCH(**) - ROBERTO LUCCHETTI(***) - ROGER J.-B. WETS (***)
Summary. - An analysis of the topology generated by the p-hausdorff distances on the hyper- space of subsets of a normed linear space. In addition, a compactness criterion is derived for the topology generated by the pointwise convergence of the distance functions (the Choquet- Wijsman topology).
1. - Preliminaries.
The hausdorff distance is the main tool to quantify the distance between the sub- sets of a metric space. This works well as long as they all lie in a bounded subspace. In many applications however, one has to deal with unbounded sets or with collections of bounded sets which are not necessarily uniformly bounded. The most telling exam- ples probably come from the study of the quantitative stability of optimization prob- lems when one needs a way to measure the distance between the epigraphs (all points that lie on or above the graphs) of extended real-valued functions. But there are many other instances, such as when considering the distance between cones, the range of unbounded operators, the domain of multifunctions, the graphs of (even bounded) operators, etc. Motivated by some of these situations, a number of authors have relied on ad hoc variants of the hausdorff distance: by renorming the space, by compactification of the space or by measuring the distance between sets restricted to bounded portions of the space. The work of two of the authors [2-5], confirmed by fur- ther studies of LEMAIRE [24-25], Az~ and PENOT [8], AzI~ [7], MOUALLIF and TOS- SINGS [30], BEER [10-11], KING [21] and MOUDAFI[31] suggests that the best ap- proach, which is universally applicable, is to generate a metric on hyperspaces of sets
(*) Entrata in Redazione il 21 giugno 1989. Indirizzo degli AA.: H. ATTOUCH: Universit~ du Languedoc, 34060 Montpellier, France; R. LUCCHETTI:University of California, Davis, CA 95616; R. J.-B. WETS: University of Califor- nia, Davis, CA 95616. (**) Visiting University of California, Davis, Spring 1988. (***) Visiting from Universit~ di Milano. Research supported by the Consiglio Nazionale delle Ricerche. (***) Research supported in part by the National Science Foundation. 304 H. ATTOUCH - R. LUCCHETTI - R. J.-B. WETS: The topology etc. by means of the p-hausdorff distances. The purpose of this paper is to describe the re- sulting topology, and to identify some of its basic properties. We shall see that in a number of important situations, this topology coincides with well-known topologies, in particular with the Fell topology (Painlev~-Kuratowski convergence) and the Cho- quet-Wijsman topology. Our results show that in such cases the ~-hausdorff can be used to calculate rates of convergence, error estimates, etc. Let us begin with a review of a few known facts about the ~-hausdorff distance [4]. Unless specifically mentioned otherwise, we always denote by (X, I1"1[) a normed lin- ear space and by d the distance function generated by the norm. For any subset C of X,
d(x, C):= inf IIx - yll yeC denotes the distance from x to C; if C = 0 set d(x, C) = ~. For .~ I> 0, ~B will be the ball of radius p centered at the origin, and for any set C, C~:= ChuB.
For C, D c X, the excess function of C on D is defined as, e(C, D) := sup d(x, D) , xeC with the (natural) convention that e = 0 if C = 0; the definition yields e = ~ if C is nonempty and D is empty. For any ,o I> 0 the p-hausdorff distance between C and D is haus~ (C, D) = sup {e(C~, D), e(D~, C)}.
Except for the ,,triangle inequality~ proved in [4, Proposition 1.2], it is immediate that for any p > 0: (a) nonnegativity: haus~ (C, D) t> 0; (b) symmetry: haus~ (C, D) = haus~ (D, C); (c) triangle inequality: for any p > d(0, C), (i = 1, 2, 3), haus~ (C], Ca) < haus3~ (C1, C2) + hauss~ (C2, C8 ) ; and moreover, (d) when C, D are dosed, haus~ (C, D) = 0 for all p > 0 if and only if C = D. A sequence of sets {S v c X, v e N} is said to converge with respect to the p-haus- dorff distances to a set S, if for all z > 0, lim haus~ (S v, S) = 0.
In view of property (d) and the fact that convergence is not affected by taking clo- sure, we shall restrict our analysis to sequences or collections of closed sets. To prove that this notion of convergence is topological requires taking special care H. ATTOUCH - R. LUCCHETTI - R. J.-B. WETS: The topology etc. 305 of the case when convergence is to the empty set. Let 5 r (or 5~(X)) denote the hyper- space of closed subsets of X, and 5~o = #\ 0, the hyperspace of nonempty closed sub- sets of X. Let us consider on ~0, a metric m~ generated by a notion of distance in- troduced in [2]: let d} (C, D):= sup jd(x, C) - d(x, D) l and (c, D) m~ (C, D) := 2, 2 -~ 4,, ~=1 I +d~(C,D) with ~ some sequence of increasing positive numbers that tend to ~. Clearly m~ is a metric on #0. Let (5~0, Z'w) be the resulting topological space. Note that the choice of the sequence { p~ } affects the distance between sets but has no effect on the topology, in fact the same uniformity is generated whatever be the sequence ~,~. Comparing z~-convergence with convergence of the ,~-hausdorff distances, we have that for any ~ > 0 and Po > max {d(0, C), d(0, D)}:
hau% (C, D) ~< d' (C, D) ~< hau%o+2o (C, D)
The first inequality is immediate, the second one is due to BEER [10, Lemma 3.1], see also [8]. From this it follows that convergence on ~0 with respect to the ~-hausdorff distances is metrizable (with metric m~w). The situation is slightly more complicated when working with ~ Then maw in- duces a topology that is strictly finer than the one that might be generated by the convergence of the p-hausdorff distances. For example, let X =R, C ~= {v), veN, and C = 0. Then for all ~ > p, haus~ (C ~, C) = 0, but for all v and p i> 0, d~ (C ~, C) = ~. The reason for this is that convergence to the empty set has different characterizations with respect to the haus~- or d~'-distances. In the first case, Lira C"= 0 means that ~-~ ~
V~ > 0, 3v~ such that Vv I> v~, (C")~ = O, whereas in the m'.-metric, it means that
3v' such that Vv ~ v', C '~ = 0. To extend the definition of d~' so as to be able to deal well with convergence to the empty set we need to compactify, and rescale the range of the distance functions. We have a way to distinguish between the distance from a (nonempty) set S to the empty set and sets that are nonempty but arbitrarily far away from S. For ~ > 0, let
d e (C, D) := sup Io(d(x, C)) - O(d(x, D)) I where 0: R+-~ [0, 1] is a scaling function, i.e., continuous, strictly increasing func- tion with 0(0) = 0 and O(x + y) <~ O(x) + o(y), for example 0(~) = (2/=) arctan(~); in a re- 306 H. ATTOUCH - R. LUCCHETTI - R. J.-B. WETS: The topology etc. lated setting this approach was already used by CORNET [15], HILDEBRAND [18], ROCKAFELLAR and WETS [32], etc. We choose specifically: 0(~) := ~(1 + ~)-1 with 0(~) = 1, and some of the argumentation that follows depends on the properties of this scaling function, but it can easily be adapted to accommodate any other scaling function. Let
m~ (C, D) = ~ 2 -P,,de~(C, D) n=l denote the associated metric on ~, and z~. the topology induced by this metric. Of course, m~. is not the only metric that induces this topology on 5~. Again, choosing an- other sequence ~, or another scaling function 0, will change the distance between sets but will have no effect on the topology. Also, replacing ll']] by any equivalent norm on X will still bring us to the same topology of The uniform structures associated with the metrics that we introduced are: ~-hau~ the uniform structure associated with the p-hausdorff distances {haus~, p > 0} on ~• ~, cf. AZE [7, Section 3], ~t'~ the uniform structure associated with the pseudo-distances {d~, ~ > 0}, ~t~ the uniform structure associated with the pseudo-distances {d~, ~ > 0}.
1.1. LEMMA. - Let ~ >i 0 and 5~ := {F e 5] F • ~B ~ ~}. On ~, the three uniform structures ~-h~, ~'~ and ~ coincide.
PROOF. - From the inequality O(x) - O(y) = x/(1 + x) - y/(1 + y) ~ (x - y)/(1 + x) 2 that holds for all x, y/> 0, for C, D e ~, we have d; (C, D) = sup ]d(x, C) - d(x, D) I
~< (1 + ~ + ~)2 sup ]O(d(x, C)) - O(d(x, D)) I = (1 + p + ~)2d~ (C, D). Ilxli ~< e This, with our earlier observations, implies that for C, D in $~,
d e (C, D) ~< d~' (C, D) ~< (1 + ~ + ~)2 de (C, D),
haus~ (C, D) ~< d~' (C, D) <~ haus~ + ~e (C, D).
And this, of course, is all what we need to conclude the proof.
As already indicated convergence with respect to the ~-hausdorff distances and convergence with respect to the d~'-distances do not coincide con ~• ~, basically be- cause the convergence to the empty set behaves differently in the two cases. The full relationship is clarified by the following theorem. H. ATTOUCH - R. LUCCHETTI - R. J.-B. WETS: The topology etc. 307
1.2. THEOREM. - On :YX ~, %L'awD "[Law=~-haus-
PROOF. - From the inequalities already used in the proof of the lemma, it follows that the only nontrivial identity that needs to be verifies is %t~w = %t~-h~u~. We begin by showing that for all ~ > 0, ~ > 0, there exist z and V such that
c G where
U~_haus(~) :: {(C,D) ;Yx ~'l haus,(C,D) < v},
U~ (~) := {(C, D) ,~• 5 I d e (C, D) < ~}.
Let us suppose to the contrary that there exist p and ~ and {(C ~ , D ~) ~• Y, v N} with d~(C~,D ~) > ~ and hausy(C~,D y) < v-1. Since d~(C~,D ~) > ~ there necessarily exists x such that ]]x]] ~
~ and this yields the exis- tence of ~ such that for all v, min {d(x, C~), d(x,D~)} < z. We use the fact that haus~(C ~, D ~) < v-1 to conclude that for all v, both C ~ and D ~ must meet the same ball. We would thus be in the setting of Lemma 1.1 and this contradicts our working hypothesis. The proof in the other direction is similar. If we assume that there exist ~ > 0, ~>0 and a sequence {(C~,D~)eh:• veN} such that haus~(C'~,D~)>~ and d~ (C ~, D ~) < v-~, we know from the lemma that this cannot occur if the sets C ~ and D y meet the same ball. Moreover, if some subsequence of the C ~ (or D y, for that matter) had nonempty intersection with some ball, then when restricting ourselves to this subsequence, from the inequality d~(C~,D~)
Rephrasing these results in terms of the metrics:
1.3. COROLLARY. - Let {S;S~,v c Y. Then, for all ~, .lim haus~(S~,S) = 0 if and only if ~m= maw(S ~, S) = O. Moreover, if S is nonempty, this occurs if and only if )ira maw(Zv, S) = O.
We shall make reference to some other notions of convergence and topologies that have been defined on Y. Painlev&Kuratowski convergence of a sequence {S~ c X, v N} to a set S means
Lim sup S V= S = Lim inf S ~, v___> o o v--~ co 308 H. ATTOUCH - R. LUCCHETTI - R. J.-B. WETS: The topology etc. with Lim inf S v= {x = lira x"lxV6 S ~ for all sufficiently large v 6 N},
Limsup S ~= {x = lira xkl x k 6S ~k for all k, {vk} a subsequence of N}.
On ~, Painlev~-Kuratowski convergence of sequences is convergence with respect to the Fell topology ~f generated by the subbase consisting of all sets of the type: (5:K,K compact), (5~6, G open), where for a set D r g~D:={FE~[FnD=O}, g:D:={Fe~:]Fv~D--/:O}. If X is locally compact and separable the Fell-topology is first countable, more- over without local compactness Painlev6-Kuratowski convergence is not topologi- cal [22, Theorem 3.3.10]. When X is a banach space, Mosco-convergence [29] of the se- quence (S v r X, ~ 6 N} to S means w- Lim sup S ~ = S = s- L!minf S ~, where w-Lim sup v--)~ is obtained by taking weak limits and s-Llminf by taking strong limits, i.e., one re- quires convergence with respect to both the strong and the weak topologies on X. The associated Mosco topology ~ will only be considered in its natural setting, namely on the subspace C c 5~ of closed convex subsets of X. The Choquet-Wijsman topology z~ is generated the pointwise convergence of the distance functions. A sequence of closed sets {F ~ c X, v 6 N} Z~w-Converges to a closed set F if there is pointwise convergence of the distance functions {d(., F ~), ~ 6 N} to d(., F). If X is separable, this topology is metrizable. An actual metric is given by the following expression Id(xn, C) - d(xn, D)i m~w(C,D):= ~ 2 -~ = 1 1 + Id(x~, C) - d(xn, D) I ' with {x~, n 6 N} a dense subset of X. LECHICKI and LEVI [23] have made a detailed study of this topology. When X is finite dimensional, then on 5~0 all these topologies and convergence no- tions coincide [33, Theorem 2.2]. In the infinite dimensional case, the relationship be- tween Painlev~-Kuratowski convergence, the ~f-topology, and the ~-topology has been thoroughly studied by FRANCAVIGLIA, LECHICKI and LEVI [16]. We also need to appel to the following known connections between these various topologies. If X is reflexive, then on C, z,~ is in general finer than Z~w [34, Proposition 2.2], [26, Theorem 4.1]; this is not true in the non-reflexive case [9, Example 2]. They coincide if (X, II'll) is reflexive and if X* has the following property: (v ~ 6 X* I llv ~N. = = 1} weak*-converges to v with Ilvll. = 1 then lira Iiv~- vii. = 0, i.e., I1" II. has the Kade5 property, see [13]. Observe that given any reflexive banach space (X, I" I), on C, Zm is invariant with respect to any other norm equivalent to I" I, and by Trojanski renorm- ing Theorem, X can be endowed with an equivalent norm H'II so that (X, I1"11)is strong- ly smoot, the norm is then fr~chet differentiable (except at the origin) and I1" II. has the Zade~ property [20]. Thus, when working with (X, I1"11), Mosco-convergence and raw- H. ATTOUCH - R. LUCCHETTI - R. J.-B. WETS: The topology etc. 309 convergence agree on e. Collecting all these facts, we can conclude that when (X, i" i) is a reflexive banach space, then there is a topology ~ on e such that (e, ~,,) induces Mosco-convergence for sequences in C; cf. also [35, Theorem 3.36] and [12, Section 3]. Finally, note that zaw convergence implies Painlev6-Kuratowski-convergence; it is also finer than Z~w even when restricted to e [4, Theorem 4.2]. Further comparisons can be found in [26], [34] and the more recent treatise of SONNTAG [35]; a few related remarks can be found later in this article. The next section is concerned with the completeness of the metric spaces (5=,ma~) and (~0, maw ). Section 3 provides compactness criteria for subset of 5= with respect to both the Taw and the ~w topologies. In Section 4, we study the connectedness proper- ties of (~, za~) and exhibit Zaw-continuous paths between the elements of ~ Separabil- ity is discussed in Section 5.
2. - Completeness.
Proposition 4.9 of [3] raised the question of the completeness of the space (#, maw ), and provided a partial answer. The issue is settled in full generality by the next theorem.
2.1. THEOREM. -Suppose (X, n'll) is complete. Then (5=,m~w) and also (5=o,m~w) are also complete.
PROOF. - Let (S v, ~ N} be a catchy sequence in (~, maw). If to every ,z > 0 there corresponds a subsequence such that (S ~)~ is eventually empty then, as already ob- served in the proof of Theorem 1.2, the sequence S v must converge to the empty set. Thus to prove completeness, it will suffice to consider the case when there exists some Po > 0 such that for all ~ > Po, (S '~),~ r 0. This condition is certainly satisfied if {S ~, v N} is a catchy sequence in (5=o, m[~w). For suppose there is some subsequence such that {(S ~)k = 0, k N}. Then for any (fixed) .$ > 0, for all k, given any x S ~k, there exists k'> k such that for all y S ~k', IlYll>0+ Ilxll. Hence for all ~ I> Ilxll, d e (S ~', S '~') I> d(0, S ~k') - d(0, S ~k)/> 0. But that would contradict the assumption that S ~ is a catchy sequence. In view of Lemma 1.1, we only need to show that {S v, ~ has a limit (in 5=0) when it is a catchy sequence for the uniformity generated by haus~. Because of the relationship between the limits of the excess functions {e((S ~)~, S), v N} and the lira- it superior of the sequence S ~ [33, Theorem 2.2], the z-limit of the sequence S '~, if it exists, must be equal to Limsup S ~ =:S. Thus we need to show that for all ~>0; ~-~ (a) e((S~)~,S) tends to 0 as ~ goes to ~, (b) e(S~,S ~) tends to 0 as v goes to ~. 310 H. ATTOUCH - R. LUCCHETTI - R. J.-B. WETS: The topology etc.
We begin with (a). Let ~ e (0, 1). From the cauchy condition we know that
V(k>0,~>0)3vk,: such that Yv,~' ~vk,.~, haus~(Sv,S~') < 2k+---7.
Now fix p, and without loss of generality, suppose that (S~)~ r 0 for all v. Let 'VO I> Y0,~+l and Xo e (S'~~ Let vl > max(vo, vl,p+l). Since haus,(S ~~ S ~) < ~2 -(~247 there exists xl e S '~' such that d(xo, xl)< ~/2. Now note that
d(xi,O)~d(Xl,X o)+d(xo,O)<~ ~ +p (and hence xl e (S ~)~ +1). By induction, we obtain for vk > max (~k-1, vk,~+ 1 ), an xk e S ~ so that d(xk, xk-1) < ~/2 k and still xk e (S~)~. The sequence {xk, k e N} c X is cauchy, with limit x. Thus x e S:/:0 (by definition of Limsup). Moreover, d(xo, x) <. ~. Summarizing, for fixed ~ > 0, ~ > 0 we found v' = v0,~+l such that for all v I> v', for all x0 e (S ~)~, there exists x e S with d(xo, x) <. ~. Hence e((S ~)~, S) <- ~. Let- ting ~ $ 0 completes the proof of (a). To prove (b), again fix ~ > 0 and p > 0. If S~ = 0 there is nothing to,prove. Other- wise let x e S~. This means that there exists (xk e (S ~): + ~, k e N} converging to x and for k sufficiently large d(x,x~)<~/2. For ,J sufficiently large, in particular Y i> VO,~+l, d(x, S ~) <- d(x, xk) + e((S ~)~ +1, S ~ ) < ~. Hence e(S~, S ~) < ~ as soon as v I> v0,~+l. This completes the proof since ~ can be cho- sen arbitrarily small. 2.2. COROLLARY. - Let {S ~, v e N} r 5~ be such that the functions {x~d(x,S~): X-.R§ = [0, ~)} converges uniformly to a function ~: X-)R+, uniformly on bounded subsets of X. Then, there exists S 5~o such that ~(x) = d(x,S). PROOF. - Take S = Limsup S v, and observe that the uniform convergence on bounded sets means that = Zaw-Lim S ~. Now, apply Theorem 2.1. If instead of Zaw-Convergence, we work with zcw-convergence, the situation is quite different. 2.3. EXAMPLE [35]. - Let H be a hilbert space (that is not finite dimensional) and (e~, v e N} an orthonormal basis. Let S ~= {y egl 3. - Compactness. We are interested in characterization of compactness for (5~, z~w ) and (5~, Zc~ ). This in turn, yields compactness conditions for functionals in the classical framework of the calculus of variations. Note it will suffice to prove sequential compactness: (5~, z~) is metrizable and thus first countable. 3.1. THEOREM. - Suppose that for all ; > 0, K (~) is a relatively compact subset of X. If6) c 5:is such that F ~ (~ implies F~ r K (~), except possibly for a finite number of sets in 0~, then (~ is %w-relatively compact. PROOF. - We need to show that from any sequence {F ~; ~ N} c 69 we can extract a sub-sequence which is raw-Convergent. Observe that if for all ~ > 0 there exists a subsequence {F v*, k N} such that (F ~)~ is eventually empty, then the F v~ converge to the empty set, see the proof of Theorem 1.2. Thus we may as welt suppose that for all v and ~, (F ~)~ is nonempty. We are going to construct this subsequence with the help of a diagonalization argument. So let us describe the k-th operation, and start with {F ~; v ~k- 1 }, a given subsequence of the initial sequence {F ~; v N}. Consid- er the sequence {Fvr~(k + 1)B; v e2;k-1 }. By assumption, this sequence is contained in the fixed relatively compact set K (k + 1) From the classical compactness results for the hausdorff metric [22, Theorem 2.3.5 and Corollary 4.2.4], we can extract a subsequence Sk c 2k-1 such that lim (F ~ • (k + 1)B) =: G k exists for the hausdorff metric. We can construct a diagonal subsequence by taking for vk, the k-th term of 2~k and define 2; = (~1, v2, ..., vk, ...}. So, for any keN, (Gk)k is a closed subset of X. We claim that for every 1 > k (G z)k = (G k)k- 312 H. ATTOUCH - R. LUCCHETTI - n. J.-B. WETS: The topology etc. The inclusion (Gl)k ~ (Gk)k follows immediately from (Gt)k = G l n kB = [lim (F" n (l + 1)B)] n kB, 'IEZ' (GZ)k ~ [!im(F~ n (k + 1)B)] n kB = (Gk)k, where limits are tken with respect to the hausdorff metric. In the opposite inclusion, for y e (Gl)k we have tYl ~ y lim(F ~ n (k + 1)B) = G k ~EZ and since tYl ~ lira (F ~)~ = G ~- 1 and that G P-1 = lim (F ~ n pB) c [!im (F" n (~ + 1) B)] n pB = (G ~)~ r F vE2~ and, of course, this yields !im e((F ~)~, F) = O. In order to prove that also lira e((F)~ ,F ~) = O, we can note that (F)~ = (G~)~ r G ~, which implies e((F)~ , F ~) <. e(G ,~, F v n (,o + 1)B) and from the definition of G ~, it follows that this last quantity tends to 0. A simpler proof can be given for the case when X is separable. In that case any se- quence has a subsequence converging in Painlev~-Kuratowski's sense (see also the proof of Theorem 3.5). But the proof given here holds with separability. Moreover the compactness condition, that in infinite dimensions is rather restrictive, is on the other hand automatically satisfied in finite dimensions. It is enough to take K (~) = ~B. Hence Theorem 3.1 provides an alternate proof of the compactness of (~, zf), when X is finite dimensional, because zaw is finer that vf; this is another reason for avoiding the use of a subsequence converging in the Painlev~-Kuratowski sense in this proof. We can use Theorem 3.1 to obtain a compactness condition in a (classical) functional setting. 3.2. THEOREM. - Let X, H be reflexive banach spaces, and ~ a continuous com- pact embedding of X into H. Given 0 >i O, and 0: R+ ---) R+ a function with bounded H. ATTOUCH - R. LUCCHETTI - R. J.-B. WETS: The topology etc. 313 (inf-) level sets, i.e., for all ~, {xl ~(x) <~ ~} is bounded, let ~ be the space of functions f defined on H with the following properties: f is proper, Isc (lower semicontinuous) convex on X, Vx X, f(x) >1 ~(][xIL~)- 011xll~, Vx H \ X, f(x) = ~. Then (epi ~):= {epi fl f 2} is z~-compact. PROOF. - A routine argument shows that the functions in ~ are lsc at every point of H. We want to apply Theorem 3.1, so let us show first that YxoeX, V(~>O,~eR)3KcHxR, compact, such that Ao := epi f n (~BH• + (Xo, ~)) r K. Let (x, f) A0. Then f ~f(x), I~1 <~ I~1 + ~ and [IxiIH<~ ]ix0I1~ + P. It follows that ~(I]xtlx) < f(x) + oHxtj~ <-I~l + ~ + o(HXoJig + ~)~. From the boundedness condition on the (inf-) level sets of ~, it follows that IIx]ix <. v, with ~, depending only on x0, a and p but not on fi The compact embedding ~ associ- ates to vBx a compact set, say K' c H. This, and the fact that ]fie ~< lal + p, allows us to conclude Ao c g'x E-(t~l + p), t~[ + P] := K, with K compact. One of the shortcomings of the condition provided by Theorem 3.1 is that it fails to yield a compactness criterion, for a sequence of unbounded sets (with nonempty inte- riors, for example), even if they are ((very close, to each other; for instance, let X be a banach space and r~:= {(x, ~)1 ~/> (1- v-1)llxll2 } c X• The next theorem makes provisions for such sequences. 3.3. THEOREM. - Suppose c {h: X--> X I h is uniformly continuous on bounded sets}, is such that is equicoercive, i.e., inf lim Iig(Y)[I = ~, .~ is relatively compact with respect to the seminorms ]. 1~, where lgi~ := sup (llg(x)N: x ~B}. 314 H. ATTOUCH - R. LUCCHETTI - R. J.-B. WETS: The topology etc. Suppose (~ r 5: is z~.-relatively compact. Then G(o~) := {cl (g(F)): g e G, r e ~, F ~e 0} w {01 if 0 e 0)}, is Vaw-relatively compact. PROOF. - Let {D ~ = cl gV (F ~), ~ 6 N} r ~((~). We need to show that this sequence has a cluster point. If Z~w-Lim F ~ = 0, then the equi-coercivity condition implies that Vaw-LimD~ = 0. Otherwise, let us suppose, passing to a subsequence if necessary, that 0 ~ F := ~a~-Lim r ~ and g = I" I~-~li~m~g". For any y, z 6 X, and g 6 G, d(g~ (y), g(z) ) <~ d(g~ (y), g(y) ) + d(g(y), g(z) ) . Hence e((g~(F~))~,g(F)) = sup inf d(g~ (y), g(z)) <~ sup inf d(g~ (y), g(z)) <~ y~FV,g~(y) egB zeF ye (F'% z~F ~< sup d(g~ (y), g(y)) + sup infr d(g(y), g(z)), y (F")~, y e (Fv)~ for some V. The existence of such a v follows from equi-coercivity. For v sufficiently large, the first term is arbitrarily small since g~ converges to g with respect to the [. I~, seminorm. And also the second term can be made arbitrarily small, simply observe that F = z~w-Lim F ~ and thus for ~ sufficiently large, to each y e (F ~)~ there corre- sponds z e F such that d(y, z) is small, and this with the uniform continuity if g (on the ball of radius v+ 1) is enough to guarantee that d(g(y),g(z)) is also (arbitrarily) small. The same arguments prove that e((g(F))~,g~(F~)) converges to 0, and hence g(F) = ~aw-Lim g~(F~). Equi-coercivity cannot be dropped without imparing the validity of the previous result. Let X be a hilbert space, ~ = (g: g(x) = v-ix, v e N}, {e~ e X; v e N} is an or- thonormal basis, and (~ = {~e~, v 6 N}. All the assumptions of the theorem are satis- fied except the equicoercivity condition. Clearly {D ~= (~-l(ve~)}, veN} is not Z~w- relatively compact. The remainder of this section is concerned with z~.-compactness. The results are of independent interest, but they also provide us with a point of comparison for the criterion for z~-compactness provided by Theorem 3.1. We begin with a review of some of the properties of Zcw-Convergence. 3.4. PROPOSITION. -Let (S;S~, ~ 6N} r and consider the following condi- tions: (ia) for all y eX, limsup d(y,S ~) <- d(y,S), v----> cc (i]a) for all x 6 X and all p > O, (x + ~B) n S --/: 0 implies that for all ~ > O, (x + + (p + D B) r~ S ~ ~ 0 for v sufficiently large, H. ATTOUCH - R. LUCCHETTI - R. J.-B. WETS: The topology etc. 315 (ib) for all y c Y, liminf d(y, S ~) >t d(y, S), (iib) for all x e X and all p > O, (x + pB) n S ~ -r 0 for ~ N' N, a subsequence, then for all ~ > O, (x + (~ + ~) B) n S -r O, (iii~) for all x e X and all p > O, (x + pB) n S ~:/= 0 for ~ e N' r N, a subsequence, then (x + pB) n S ~ O. We have that (i~) <:> (ii~), (ib) <:> (iib) ~ (iiib). In particular S = ~-Lim S" if and only if (ii~) and (lib) hold. v.--e, cc PROOF. - This follows all rather directly from the definition of d(y, D) for a set D ~ the details can be patterned after the proof of [16, Proposition 2.1]. 3.5. THEOREM. - Let (X, II'll) be separable and (~ r ~. Then the following are equivalent: -- ~ is sequentially Z~w-relatively compact, --for all xeX, all ~>0, ~>0 and all {F ~,v such that F ~(~(x+ + ,zB) r Ofor all v there exists K~, a relatively compact subset of X, and a subse- quence N' c N, such that K~ r x + (p + ~) B and K~ n F ~ r 0 for all ~ N'. PROOF. - Since X is separable (~, Zcw) is metrizable it will be sufficient to work with sequential compactness. Separability also means that every sequence (F", ~ e iV} r 0~, passing to a subsequence if necessary, admits a Painlev~-Kuratowski limit, say F, cf. [1, Theorem 2.23]. This convergence implies condition (is)of Proposi- tion 3.4, see e.g. [16, Theorem 2.2]. Next, passing again to a subsequence whenever necessary, let us assume that F" • (x + ,oB) r 0 for all v e N. The second condition in the statement implies that for all v e N, V~ > 0, 3x" ~ F ~ (~ (x + (p + ~) B) and thus ~ := lim x ~ with ~ F n (x + (~ + ~) B). This means that condition (iib) of Proposition 3.4 is also satisfied and F = zcw-Lim F v, i.e. 0) is few-relatively com- pact. Now, let us suppose that (~ is ~cw-relatively compact, and consider a sequence {F v , ~ N} r 6~ with Zcw-Lim F ~ = F ~, and such that for all v N, F ~ n (x + pB) r 0 for some x e X and p > 0. This implies that lira inf d(x, F ~) >I d(x, F) 316 H. ATTOUCH - R. LUCCHETTI - R. J.-B. WETS: The topology etc. and Vy e F, 3y ~ F '~ such that y = lim y". Since we also have that d(x, F)<<.~ which implies that for all ~ > 0 there exists y F n (,~ + DB, we choose the compact set, K~:= {y~,veN} c x+(~+ ~)B, i.e., the second condition is also satisfied. With even more structure, ~-compactness implies an even stronger condition than that of Theorem 3.5. Recall that a strongly smooth normed linear space, is one that has a Fr~chet-differentiable norm, except at the origin. An E-space is a reflexive banach space such that the norm has the Kade~ property, cf. [20]. 3.6. PROPOSITION. - Let X be a E-space such that the dual norm [[. ][. has the Kadee property, and (~ c e, a subset of the hyperspace of closed convex subsets of X. If ~ is sequentially r~w-relatively compact, then for all (x X, p >f O, {C ~, ~ N} c (~) such that C~ n (x + pB) # O for all yeN, there exists a relatively compact set K c ~B such that Cv r~ K r O for all ,~ e N. PROOF. - Let (C", ,J N} r O) be a sequence that satisfies the assumptions. Let x" = prjc~x and let K = (x ~, v Clearly K c x +#B. Passing to a subsequence if necessary, by relative compactness of 0~, there exists C := zcw-Lim C" with C C. Due to the strong smoothness of the norm this means that the sequence Mosco-converges to C. This, together with the fact that X is an E-space, implies that the projections converge strongly (to prjc x), cf. [26]. The Kade5 property of II'[I, is required for (zcw-convergence~Mosco-conver- gence) [12] and the E-space structure is needed to obtain the strong convergence of the projections, see [26], [27]. However, uniqueness of the projections is not needed, so there is no need for strict convexity. If H is a hilbert space, {e,,, ~ e N} an orthonormal basis, C = {(1 + v-1 ) e~, v N} and C" = C u {e,, }. Then 6~ = {C; C~: v N} is Zcw-compact and thus satisfies the con- dition of Theorem 3.5. However, it does not satisfy the condition of Proposition 3.6. Similar examples show that conditions (iia) and (iib) of Proposition 3.4 cannot be mod- ified to conditions with, = 0. This example also highlights the role played by convexi- ty in Proposition 3.6. Finally, we observe that if zl crs are two topologies on a space Y and (Y, ~2) is hausdorff, then (Y, ~1 ) compact forces zl = ~2. Hence the last proposition allows us to H. ATTOUCH - R. LUCCHETTI - R. J.-B. WETS: The topology etc. 317 identify subsets of C, on which the z~-, ~.-topologies and that induced by Mosco-con- vergence coincide. In particular, we note that Theorem 3.2 is an extension of[4, The- orem 4.7]. 4. - Connectedness. 4.1. THEOREM. - (if, ~aw) is path-connected. PROOF. - Let 0 r F 5, and .~: [0, 1] --~ ~ such that F+(e/(1-a))B force[0,1), ~b(~) = { if ~ = 1. Note that ~b is continuous (even with respect to the hausdorff metric) for all ~ < 1. When ~ = 1, for any ~ > 0, choose % sufficiently close to 0 so that ~b(~s ) D 9B. Hence, for all 1 > a > %, e(X~ ,~(~)) = O, and hence hauss (@(a), X) = 0 for ~ sufficiently close to 1. Thus any nonempty set F is connected to all elements of 5 ~, except possibly the empty set. To show that 0 is not an isolated point, we simply redefine ~b0(a)={~\(a/(1-a))(intB) forif~=l.~e[0,1), and show that it is continuous by a similar argument. Connectedness of C can be proved by using (,shortest- paths. Let C1, C2 be two nonempty closed convex subsets of X. Let 9 > 0 be such that (C1)s and (C2)~ are nonempty, and consider the three following paths IClr~(~+(1-~)~-I)B for~e(0,1], ~1(~) = [C1 if ~ = 0; ~2 (a) = (1 - ~)(C1 )s + a(C2)~ for ~ e [0, 1] ; IC2c~(,~+(~(1-~)-I))B for~6[0,1), ~8 (a) = [C2 ff a = 1. The proof of the continuity of ~ and ~3 relies on [4, Proposition 1.4]. 5. - Separability. 5.1. THEOREM. - (5~, Yaw) is separable if and only if X is finite dimensional. PROOF. - We already know that in finite dimension %w -- ~f which is known to be metrizable. Furthermore, see e.g. [1, Proposition 2.77], zf has a countable base, and 318 H. ATTOUCH - R. LUCCHETTI - R. J.-B. WETS: The topology etc. in metric spaces this is equivalent to separability. This takes care of the ,(if)~ part. For the ,,only ft, part, if X is infinite dimensional, there exists a sequence {x ~ e B, v e N}, such that Ix~-xr I> 1 whenever v~:~'[18, Lemma 1.12.2]. Let 0) = {S IS= = [J x~,2 e 2 N } r ~. Observe that 6~ is not countable, and ifS~,S~ ~ (D, S~ :/:$2 then vE,~ hi ($1, $2 ) ~ 1. It follows there is not countable dense subset of 6~, and thus the space (~, v~) cannot be separable. " If X is a hilbert space, we can take for {xVe B, ,J e N} on orthonormal basis {e~, ~ e N} and take for (~ = {co SI S = [J ev, ~ e 2N }. From this it follows that (C, v~w) vEZ is not separable, and in turn this means that the space of convex functions defined on X, when X is infinite dimensional, it not separable when equipped with the topology induced by ~w-topology on the epigraphs. The situation is quite different when 5~is equipped with the Z~w-topology. Suppose X is a separable banach space, then to obtain the separability of the hyperspace (~, Z~w) it suffices to observe that z~ is coarser than the Vietoris topology [16, Propo- sitions 2.1 and 2.3] and that 5 ~ endowed with the Vietoris topology is separable [28, Proposition 4.5.1]. A more constructive approach is the following: let D be a dense subset of X, then for all ~ > 0 and F e ~, there is DE a finite subset of D such that sup {Id(x,F) - d(x, DF)l: cardM finite} < ~. xeMcD From the definition of m~w this amounts to saying that the family of finite subsets of D is dense in (5~, Zcw); for details see [17, p. 1-6]. Acknowledgment. We are very grateful to Professors D. Azs C. CASTAING and G. BEER for a careful reading of an earlier manuscript. Their welcome comments have helped the authors avoid a number of possible pitfalls. REFERENCES [1] H. ATTOUCtt, Variational Converges for Functions and Operators, Pitman, London (1984), Applicable Mathematics Series. [2] H. ATTOUCH - R. J.-B. WETS, Isometries for the Legendre-Frenchel transform, Transac- tions of the American Mathematical Society, 296 (1986), pp. 33-60. [3] H. ATTOUCH - R. J.-B. WETS, Quantitative stability of variational systems: I. 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