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Hausdorff Distance (*) 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.
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