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Acta Mathematica Sinica, English Series (Springer) Subdifferentials of Distance Functions, Approximations and Enlargements
Subdifferentials of Distance Functions, Approximations and Enlargements
JeanPaul Penot, Robert RatsimahaloHow much do you like this book?
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Volume:
23
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english
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14
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Acta Mathematica Sinica
DOI:
10.1007/s1011400507636
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March, 2007
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Acta Mathematica Sinica, English Series Mar., 2007, Vol. 23, No. 3, pp. 507–520 Published online: Apr. 28, 2006 DOI: 10.1007/s1011400507636 Http://www.ActaMath.com Subdiﬀerentials of Distance Functions, Approximations and Enlargements JeanPaul PENOT Department of Mathematics, Faculty of Sciences, av. de l’Université, BP 1155, 64013 PAU Cédex, France Email: jeanpaul.penot@univpau.fr Robert RATSIMAHALO Business U nit : Bank and Financials, SODIFRANCE, 104 rue Castagnary, 75015 Paris, France Email: rratsimahalo@sodifrance.fr Abstract In this work, we study some subdiﬀerentials of the distance function to a nonempty nonconvex closed subset of a general Banach space. We relate them to the normal cone of the enlargements of the set which can be considered as regularizations of the set. Keywords Distance function, Normal cone, Regularization, Subdiﬀerential MR(2000) Subject Classification 49J52, 49J50, 58C20, 46B20, 46N10 1 Introduction A large literature has been devoted to the study of the diﬀerentiability of the distance function dC := d(·, C) to a closed nonempty subset C of a Banach space X. This topic has many applications in various domains such as mechanics, economy, best approximation, nonsmooth analysis, Hamilton–Jacobi equations, invariance and viability of sets for dynamical systems. In fact, distance functions are special cases of value (or performance) functions and play a key role in nonsmooth analysis, in several works normal cones and subdiﬀerentials are constructed by means of such functions. It is the aim of the present work to ﬁnd explicit formulae for the generalized subdiﬀerentials of dC . Several such subdiﬀerentials have been considered elsewhere: the Clarke subdiﬀerential ([1, 2, 3, 4, 5, 6, 7, 8, 9],. . . ), the Fréchet subdiﬀerential ([1, 2, 3, 4, 10, 11]), the proximal subdiﬀerential ([7, 8, 9]), the moderate subdiﬀerential of Michel–Penot ([4, 12, 13],. . . ). Therefore, here we deal only with the Fréchet subdiﬀerential, the Hadamard (or Dini–Hadamard or contingent or ; directional) subdiﬀerential and the limiting subdiﬀerential which are also basic notions of nonsmooth analysis. The ﬁrst two notions generalize the derivatives in the sense of Fréchet and Hadamard, respectively (the latter being equivalent to the Gâteaux derivative for a Lipschitzian function such as dC ). They are closely connected with geometrical notions such as normal cones and tangent cones. These appears that several results are in sharp contrast with the ones for the Clarke subdiﬀerential (see the remark following Proposition 15). On the other hand, after the writing of the ﬁrst versions of the present paper, we learned from [13], [14] the equivalence of several properties bearing on the distance function for various subdiﬀerentials. Received June 1, 2004, Accepted September 21, 2004 The research of this work has been partly carried out when the second author was visiting the Department of Mathematics at the Chinese University of HongKong, at the invitation of Prof. K.F. NG. The visit was made possible by ﬁnancial supports from the Research Council of HongKong, and from the General Consulate of France Penot J. P. and Ratsimahalo R. 508 Since we are working in a general Banach space without any inner product, some tools from the geometry of Banach spaces are required. They are recalled in a preliminary section along with some results about approximations of sets which play an important role in the following. In Section 3 we deal with the subdiﬀerentials of the distance function dC to C at a point w ∈ C, while Section 4 is devoted to the case of a point outside C. While several of our results are similar to results in [5], [9], they diﬀer from these references since we use Hadamard or Fréchet subdiﬀerentials, instead of Clarke subdiﬀerentials or proximal subdiﬀerentials, and since we are working in a general Banach space. Several results given in Section 4 rely on the nonemptiness of the set of best approximations to a closed subset. The formulae obtained in Section 4 are more explicit when one regularizes the closed subset C as in [5], [9], [15] via a technique of enlargement. As the enlargement studied in [5] is not necessarily closed in an inﬁnitedimensional space, we adopt in Section 5 another enlargement, which has been studied in [7], [9] and [15], in particular in the case X is Hilbertian. The regularity conditions we assume for the subdiﬀerential formulae of the distance function are quite diﬀerent from the Clarke regularity condition which has been much used (see for instance [5], [6], [3],. . . ). Our main result, Corollary 32, relates the normal cone to the enlargement to the Fréchet subdiﬀerential of the distance function, generalizing [9]. Our method is inspired by arguments in [3, 16, 17, 18]. Section 6 is devoted to the special case of a convex set C; then the preceding results take a more striking form. The results we obtain explain the regularizing eﬀect of taking enlargements when some favorable assumptions are available. 2 Preliminaries Throughout this work, X is a Banach space with topological dual X ∗ and C is a nonempty closed subset of X. We denote by BX the closed unit ball and by SX the unit sphere of X. We set k(x) = 12 x 2 and n(x) = x. The distance function dC is given for w ∈ X by dC (w) := d (w, C) := inf x − w = inf (x − w + ιC (x)) = (nιC ) (w) , x∈C x∈X where ιC is the indicator function of C taking the value 0 on C and +∞ elsewhere and where is the infconvolution operator. In this section, we recall some concepts and results needed for our study. The duality mapping (or rather multimapping) J (resp. S) is given by J (0) = {0} (resp. S (0) = BX ∗ ), and for x = 0 by J(x) = {x∗ ∈ X ∗ : x∗ , x = x2 , x∗ = x}, resp. S(x) = x−1 J (x) . In fact, J = ∂k (resp. S = ∂n), where ∂ is the Fenchel subdiﬀerential, i.e. the usual subdiﬀerential in the sense of convex analysis, n (x) := x, and k (x) := 12 x2 . Since k (resp. n) is convex continuous, the radial derivative of k (resp. n) (which coincides with its directional derivative) exists. Then, we set 1 x, y+ := k (x, y) := limt→0+ (x + ty2 − x2 ), 2t 1 (resp. [x, y]+ := n (x, y) := limt→0+ (x + ty − x)). t The bracket ., .+ (resp. [., .]+ ) is called the semiinner product (resp. the normalized semiinner product) (see [19]). Moreover, the subdiﬀerential J = ∂k (resp. S = ∂n) of k (resp. of n) is related to the semiinner product through the relation x, y+ = max {x∗ , y : x∗ ∈ J (x)} (resp. [x, y]+ = max {x∗ , y : x∗ ∈ S (x)}), (see for example [20], [21], [22]), which is a special case of the relation f (x, y) = max {x∗ , y : x∗ ∈ ∂f (x)}, for f a continuous convex function. For a (nonempty) subset C of X and x ∈ cl(C), the tangent cone T (C, x) to C at x is given by T (C, x) := lim supt→0+ t−1 (C − x). When C is convex, its tangent cone is the closure of the radial tangent cone T r (C, x) := R+ (C − x). The normal cone to C at x is the polar cone of the tangent cone: ◦ N (C, x) := (T (C, x)) = {x∗ ∈ X ∗ : x∗ , x ≤ 0, ∀ x ∈ T (C, x)} . Distance Functions 509 For C convex, N (C, x) = (T (C, x))◦ = {x∗ ∈ X ∗ : x∗ , y − x ≤ 0, ∀ y ∈ C} . Given a realvalued function f, ﬁnite at x ∈ X, its Hadamard (or contingent or lower directional) derivative at x in the direction y is deﬁned by f (x + tz) − f (x) f (x, y) := lim inf (t,z)→(0+ ,y) . t The function f is said to be directionally diﬀerentiable at x in the direction y if the limit inf is a limit. The Hadamard (or contingent or directional) subdiﬀerential ∂f of f is deﬁned by ∂f (x) = {x∗ ∈ X ∗ : x∗ , y ≤ f (x, y), ∀ y ∈ X} f (x + tu) − f (x) − x∗ , tu ∗ ∗ = x ∈ X : ∀ v ∈ X, lim inf ≥0 . t (t,u)→(0+ ,v) The Fréchet subdiﬀerential of f at x is the subset ∂ − f (x) of ∂f (x) given by f (x + u) − f (x) − x∗ , u ∂ − f (x) = x∗ ∈ X ∗ : lim inf ≥0 . u↓0 u The fact that for a convex function f , ∂f = ∂ − f coincides with the Fenchel–Moreau subdiﬀerential will be useful in the following. In particular, when C is convex, ∂dC = ∂ − dC is nonempty as dC is convex and continuous. While N (C, x) = ∂ιC (x), the Fréchet normal cone N − (C, x) (or ﬁrm normal cone) to C at x ∈ C is ∂ − ιC (x), hence is given by x∗ , u − x N − (C, x) = x∗ ∈ X ∗ : lim sup ≤0 . u(∈C\{x})→x u − x Definition 1 ([23, 24, 25]) A set C is said to be approximated by D (in the Maurer–Zowe sense) at x ∈ clC ∩ clD, if there exists a mapping h from C into D which is tangent to the identity IX of X at x with respect to C, i.e. such that limu(∈C\{x})→x h(u)−u u−x = 0. In [26, 27] this notion is compared with a number of other concepts and some suﬃcient conditions ensuring this approximation assumption are provided; they require the following deﬁnitions. The ﬁrst one is a local version of a concept used in [28, 29, 30, 31, 32]. Definition 2 The set C is said to be pseudoconvex at x if C − x ⊂ coT (C, x) ; it is locally pseudoconvex (resp. locally quasistarshaped) at x if for some open subset U containing 0 one has (C − x) ∩ U ⊂ coT (C, x) (resp. (C − x) ∩ U ⊂ T (C, x)). Obviously every convex set is quasistarshaped at any point, and every locally quasistarshaped set C at x is locally pseudoconvex at x. Simple examples show that the converses do not hold in general. The following result is immediate: Proposition 3 If a closed subset C is locally quasistarshaped at x ∈ C, then it is approximated by x + T (C, x) at x. If a closed subset C is a submanifold of class C 1 (with or without boundary) then it is approximated by x + T (C, x) at x, for any x ∈ C. Our second criterion for the approximation property involves a compactness property. Definition 4 ([25], [33]) A set C is said to be tangentially compact at x ∈ cl C, if for any sequence (xn )n ∈ C converging to x, with xn = x for each n, the sequence xxnn −x −x n has a converging subsequence. Clearly if X is ﬁnite dimensional, any subset C of X is tangentially compact at each x ∈ C. 1/2 shows that there are interesting inﬁnite The example of the epigraph of the function x → x dimensional examples. Proposition 5 ([26], [27]) If a closed subset C is tangentially compact at x ∈ C, then it is approximated by x + T (C, x) at x. Penot J. P. and Ratsimahalo R. 510 3 Study at a Point of C In this section, we study the subdiﬀerentiability of the distance function at a point x ∈ C. The following lemma is classical (see for instance [3], [34], [35] for relation (1) and [5] for relation (2)): Lemma 6 Let C be a nonempty closed subset of X and x ∈ C. Then (1) T (C, x) = {u ∈ X : dC (x, u) = 0} , dC (x, .) ≤ dT (C,x) (.) . (2) One easily deduces the following result from that statement: Proposition 7 Let C be a nonempty closed subset of X and x ∈ C. Then ∂dC (x) ⊂ BX ∗ ∩ N (C, x). Proof Let x∗ ∈ ∂dC (x). Since dC is Lipschitzian with rate 1, one has x∗ ≤ 1. Moreover, by relation (1), for each u ∈ T (C, x) one has x∗ , u ≤ dC (x, u) = 0, thus x∗ ∈ BX ∗ ∩ N (C, x). A more precise relation can be given when the Hadamard subdiﬀerential ∂ is replaced by the Fréchet subdiﬀerential ∂ − and the normal cone N (C, x) is replaced by the Fréchet normal cone N − (C, x) . Proposition 8 Let C be a nonempty closed subset of X and x ∈ C. Then (3) ∂ − dC (x) = BX ∗ ∩ N − (C, x) . −1 − Moreover, for any v ∈ J (N (C, x)) , the vector v is approximately metrically normal to C at x in the sense that limt→0+ 1t dC (x + tv) = v . Proof The ﬁrst assertion is proved in ([10, Lemma 3]). The second one is proved in [36] when X is a Hilbert space, but here we make no assumption on X. Given t > 0 and v ∈ X such that J(v) ∩ N − (C, x) contains some v ∗ , let xt ∈ C be such that rt := t−1 xt − x − tv ≤ t−1 dC (x+tv)+εt with (εt ) → 0 as t → 0+ . Then, by the deﬁnition of J(v) as the subdiﬀerential 2 of 12 · at v, we have 2 2 t−2 dC (x + tv)2 ≥ t−1 (x − xt ) + v − εt ≥ t−1 (x − xt ) + v − 2εt rt + ε2t 2 ≥ v + 2v ∗ , t−1 (x − xt ) − 2εt rt + ε2t . Now rt ≤ v+εt and xt − x ≤ t v+trt so that t−1 xt − x is bounded. Taking a sequence (tn ) → 0+ such that lim inf v ∗ , t−1 (x − xt ) = limv ∗ , t−1 n (x − xtn ), n t→0 −1 and such that (n ) := tn xtn − x has a limit ∈ R+ , the relation v ∗ ∈ N − (C, x) yields lim inf v ∗ , t−1 (x − xt ) = limv ∗ , xtn − x−1 (x − xtn )n ≥ 0, t→0 n when n = 0 for n large enough, and also when n = 0 for inﬁnitely many n ∈ N. It follows that lim inf t→0 t−2 dC (x + tv)2 ≥ v2 . Since t−1 dC (x + tv) ≤ v , we get t−1 dC (x + tv) → v . Our aim now is to give suﬃcient assumptions ensuring that a relation similar to (3) holds for the directional notions. Proposition 9 If C is approximated by x + co(T (C, x)) at x ∈ C, in particular if C is pseudoconvex at x, then ∂dC (x) = BX ∗ ∩ N (C, x) = BX ∗ ∩ N − (C, x) = ∂ − dC (x). Proof When C is approximated by x + T (C, x) at x, one has N − (C, x) = N (C, x) by [26, 27] and an easy adaptation of the proof shows that the same relation holds when C is approximated by x + co(T (C, x)) at x ∈ C. Thus the result is a consequence of Proposition 8: ∂dC (x) ⊂ BX ∗ ∩ N (C, x) = BX ∗ ∩ N − (C, x) = ∂ − dC (x) ⊂ ∂dC (x). The following result completes Proposition 7 and clariﬁes the links between the notions introduced previously. Lemma 10 Let C be a nonempty closed subset of X and let x ∈ C. Then (i) ⇒ (ii) ⇒ (iii): (i) C is approximated by x + T (C, x) at x; (ii) dC (x, .) = dT (C,x) (.) ; Distance Functions 511 (iii) ∂dC (x) = BX ∗ ∩ N (C, x). Proof (i)⇒(ii) Let u ∈ X and let r > dC (x, u). Then there exists a sequence (tn ) → 0+ such that dC (x + tn u) < tn r. By deﬁnition of the distance function, there exists a sequence (xn ) in C such that xn − (x + tn u) < tn r. Setting vn := t−1 n (xn − x), this inequality shows that the sequence (vn ) is bounded. As the closed subset C is approximated by x + T (C, x) at x, there exists a mapping h from C into x+T (C, x) which is tangent to the identity IX at x with respect to C. Thus, since the sequence (vn ) is bounded, h (x + tn vn ) − x − tn vn = tn εn , where (εn ) → 0+ . Setting zn := t−1 n (h (x + tn vn ) − x) , we have zn ∈ T (C, x) and zn − vn = εn , thus u − zn < r + εn . Thus we get dT (C,x) (u) ≤ r, hence dT (C,x) (u) ≤ dC (x, u) and, in view of relation (2), (ii) holds. (ii)⇒(iii) Assume that dC (x, .) = dT (C,x) (.). Let x∗ ∈ BX ∗ ∩ N (C, x). For any u ∈ X and any v ∈ T (C, x), we have x∗ , u ≤ x∗ , u − v ≤ x∗ u − v ≤ u − v , hence, taking the inﬁmum over v ∈ T (C, x),x∗ , u ≤ dT (C,x) (u) = dC (x, u) . Thus x∗ ∈ ∂dC (x). Then Proposition 7 yields ∂dC (x) = BX ∗ ∩ N (C, x) . In [1] it is shown that if (ii) holds for any closed subset of X and any x ∈ X, then X is ﬁnite dimensional. The following counterexample shows that (iii) is a condition strictly weaker than (ii). Example (See [1], [17]) Let X be an inﬁnitedimensional Banach space. There exist a sequence (en )n∈N in the unit sphere of X and δ> 0 such that, for m = n, one has en − em > δ. Let C be the compact set given by C := {0}∪ 4−n e0 + 4−1 en : n = 1, 2, . . . . Then for x = 0 one has T (C, 0) = {0} (see [1]) and 1 dC (0, e0 ) ≤ < e0 = dT (C,0) (e0 ) . 4 However equality (iii) of Theorem 10 still holds since, for any u in X, one has dC (0, u) = u (see [1]). One also deduces from Theorem 10 that the set C is not approximated by x + T (C, x) at x. We have thus shown that the approximation condition plays a key role in the study of the Hadamard subdiﬀerential of the distance function (at a point x of the closed subset C). This condition is even more geometrically signiﬁcant than condition (ii). In view of Lemma 5, we get a slight extension of a known result [5]. Corollary 11 Let C be a nonempty closed subset of a Banach space X and let x ∈ C. If C is tangentially compact at x, in particular if X is ﬁnite dimensional, then dC (x, ·) = dT (C,x) (·). 4 Study at a Point Outside the Subset C In this section, we study the subdiﬀerentiability and the diﬀerentiability of the distance function dC at some point w ∈ X\C, where C is a nonempty closed subset of X. The following result is known when X is reﬂexive ([4]) or when dC is Fréchet diﬀerentiable ([18]). We provide a proof for the reader’s convenience, which does not rely on these assumptions; see also [2]. Lemma 12 Let C be a nonempty closed subset of X and w ∈ X \ C. Then ∂dC (w) ⊂ BX ∗ , ∂(−dC )(w) ⊂ BX ∗ and ∂ − dC (w) ⊂ SX ∗ . Thus dC is a supersolution and a subsolution (in the sense of Fréchet viscosity solutions) of the Hamilton–Jacobi equation H(Du(x)) = 1, x ∈ Ω := X\C, u(x) = 0, x ∈ ∂Ω, ∗ ∗ with H(x ) := x for x∗ ∈ X ∗ . Proof The ﬁrst inclusions are consequences of the fact that dC is Lipschitzian with rate 1. Now, let w∗ ∈ ∂ − dC (w) and let (xn ) be a sequence of C such that (xn − w)n converges to dC (w). Then, there exists a sequence (tn ) in (0, 1) with limit 0 such that xn − w ≤ dC (w)+t2n . Penot J. P. and Ratsimahalo R. 512 Since w∗ ∈ ∂ − dC (w), there exists a sequence (εn ) → 0+ such that dC (w + tn (xn − w)) − dC (w) − w∗ , tn (xn − w) ≥ −εn tn xn − w . Then, by the choice of (tn ), we have dC (w + tn (xn − w)) ≤ w + tn (xn − w) − xn = (1 − tn ) w − xn ≤ dC (w)+t2n −tn w − xn and we obtain −t2n + tn w − xn w∗ , tn (w − xn ) tn ≥ − εn ≥ 1 − − εn . w∗ ≥ tn w − xn tn w − xn w − xn Taking n → +∞, it follows that w∗ ∈ SX ∗ . The inclusion ∂ − dC (w) ⊂ SX ∗ yields the following property proved in [4] under the assumptions that X is smooth and reﬂexive (which imply strict convexity of X ∗ by [37 p. 43]). Proposition 13 If the dual space X ∗ is strictly convex, then, for any nonempty closed subset C of X and any w ∈ X \ C, ∂ − dC (w) is at most a singleton. If, moreover, C is convex then ∂ − dC (w) is a singleton and dC is Gâteauxdiﬀerentiable at w. Proof As ∂ − dC (w) is convex, the ﬁrst assertion is a consequence of the fact that any convex subset of the unit sphere of a strictly convex Banach space is at most a singleton. The second assertion stems from the fact that when C is convex, dC is convex continuous, hence subdiﬀerentiable (see also [37], [38]). The conclusion of Lemma 12 can be improved by assuming the existence of best approximations; for w ∈ X, we denote by P (C, w) the metric projection of w onto C given by P (C, w) := {x ∈ X : dC (w) := x − w} . We ﬁrst extend relation (2) to the case where the point w is oﬀ C. Lemma 14 Let C be a nonempty closed subset of X. Let w ∈ X and let x ∈ P (C, w). Then (4) dC (w, .) ≤ dC (x, .) ≤ dT (C,x) (.) , ∂ − dC (w) ⊂ ∂ − dC (x) = BX ∗ ∩ N (C, x) , − ∂dC (w) ⊂ ∂dC (x) ⊂ BX ∗ ∩ N (C, x) . (5) − Proof The inclusion ∂ dC (w) ⊂ ∂ dC (x) is a consequence of the relations w∗ , v − o(v) ≤ dC (w + v) − dC (w) ≤ dC (x + v) + w − x − dC (w) = dC (x + v) − dC (x), for any w∗ ∈ ∂ − dC (w). In order to show that dC (w, u) ≤ dC (x, u) for any u ∈ X, we take v := tn u in the last inequalities above, where the sequence (tn ) → 0+ is such that dC (x, u) = limn t−1 n dC (x + tn u). Then, dividing by tn and taking limits, we get −1 dC (w, u) ≤ lim inf t−1 n (dC (w + tn u) − dC (w)) ≤ lim tn (dC (x + tn u) − dC (x)) = dC (x, u) . n n The relation ∂dC (w) ⊂ ∂dC (x) ensues. The last inclusion has been shown in Lemma 7. Proposition 15 Then Let C be a nonempty closed subset of X. Let w ∈ X \ C with P (C, w) = ∅. ∂dC (w) ⊂ x∈P (C,w) (S(w − x) ∩ N (C, x)) , (6) 1 2 d (w) ⊂ (J(w − x) ∩ N (C, x)) . (7) x∈P (C,w) 2 C If, moreover, for some x ∈ P (C, w), dC (w, .) = dT (C,x) (.), the above inclusions are equalities and ∂dC (w) = S(w − x) ∩ N (C, x) . ∂ Proof It suﬃces to prove the ﬁrst inclusion since, for x ∈ P (C, w), one has 1 2 d (w, .) = w − x dC (w, .) J(w − x) = w − x S(w − x). 2 C Let w∗ ∈ ∂dC (w). Since the norm is convex, for each v ∈ X, one has 1 1 w∗ , v ≤ lim inf (dC (w + tv) − w − x) ≤ limt→0+ (w − x + tv − w − x) ≤ [w − x, v]+ . t→0+ t t Hence w∗ ∈ x∈P (C,w) S(w − x). Taking relation (5) into account, we get relation (6). Distance Functions 513 Suppose now that dC (w, .) = dT (C,x) (.) , for some x ∈ P (C, w). Let w∗ ∈ S(w − x) ∩ N (C, x). Then, for any u ∈ X, one has inf {w∗ , v + w∗ , u − v : v ∈ T (C, x)} ≤ w∗ inf {u − v : v ∈ T (C, x)} , whence (as w∗ = 1), w∗ , u ≤ dT (C,x) (u) = dC (w, u) ; thus w∗ ∈ ∂dC (w). The assumption dC (w, .) = dT (C,x) (.) seems diﬃcult to check. However, if the closed subset C is tangentially regular, in the sense that its Clarke tangent cone coincides with its tangent cone, and if its distance function is regular, in the sense that its Clarke derivative coincides with its contingent derivative, it is known (see for instance [5]) that equality holds. In particular, it is satisﬁed if C is convex. Remark Relation (6) is in sharp contrast with what occurs for the Clarke subdiﬀerential ∂ 0 dC (w) since, by [4] Lemma 1, it satisﬁes the relation ∂ 0 dC (w) ∩ S(w − x) = ∅. In particular, when the norm is Gâteaux diﬀerentiable at w − x, one has ∂dC (w) ⊂ S(w − x) instead of S(w − x) ⊂ ∂ 0 dC (w). When the norm is Gâteaux diﬀerentiable at w − x, for each x ∈ P (C, w), one has ∂dC (w) ⊂ x∈P (C,w) S(w − x), x∈P (C,w) S(w − x) ⊂ ∂ 0 dC (w). Using the fact that a normed vector space is strictly convex if, and only if, the multimapping J = ∂( 12 .2 ) is injective, i.e. for any x, y ∈ X the sets J (x), J(y) are disjoint when x = y (see [39]), one gets assertion (a) of the following corollary given in [40] and [18] under a diﬀerentiability assumption. Assertion (b) follows from the relation dC (w + tu) ≤ w + tu − x for any u ∈ X, t ≥ 0, x ∈ P (C, w). Corollary 16 Let C be a nonempty closed subset of X and let w ∈ X\C. Suppose ∂dC (w) is nonempty. (a) If X strictly convex then P (C, w) is at most a singleton set; (b) If the norm of X is Gâteaux diﬀerentiable at x − w for some x ∈ P (C, w) then dC is Gâteaux diﬀerentiable at w. The following corollary provides several ways of writing the derivative of the distance function: Corollary 17 Let C be a nonempty closed subset of X. Assume the norm of X is Gâteaux diﬀerentiable on X \ {0}. Suppose dC is Gâteaux diﬀerentiable at w ∈ X\C and P (C, w) is nonempty. Then one has the following equivalent relations : {dC (w)} = x∈P (C,w) {S(w − x)} ; = inf [w − x, v]+ : x ∈ P (C, w) , ∀ v ∈ X; 1 (d2C ) (w) = x∈P (C,w) {J(w − x)} ; 2 −1 1 2 (d ) (w) = S −1 (dC (w)). w − P (C, w) ⊂ J 2 C w−x for w ∈ X \C and x ∈ P (C, w) In particular, if X is a Hilbert space, one has dC (w) = w−x p when dC is Gâteaux diﬀerentiable at w. If X = L with 1 < p < +∞, w−x 1−p p−2 dC (w) = w − xLp w − x (w − x) = . w − xLp Let us observe that the preceding corollary describes the gradient of the distance function in the sense of Golomb–Tapia ([41]; see also [42]). The nonemptiness of the gradient is not obvious in the nonreﬂexive case, even when the function is continuously diﬀerentiable. Let us point out that under the assumptions of Corollary 17, the duality mapping J is singlevalued but not necessarily injective. Then if the space X is not strictly convex, P (C, .) may be multivalued. However, J takes the same values at any w − x, for x ∈ P (C, w). dC (w), v Penot J. P. and Ratsimahalo R. 514 Proposition 15 may provide negative results concerning the subdiﬀerentiability or the differentiability of the distance function dC . Corollary 18 If dC is subdiﬀerentiable at w with d (0, ∂dC (w)) < 1, then P (C, w) is empty. The following example illustrates this negative fact: 1 Example ([18]) Let X = en : n ∈ N and C be the closed subset given by C = 1 + 2 n \ {0} , where {en }n∈N\{0} is the canonical basis of 2 : dC is Gâteaux diﬀerentiable at 0 with dC (0) = 0. As expected, 0 has no best approximation in C. On the other hand, one has the following criterion for the existence of best approximations. As in [18] we say that w∗ ∈ X ∗ strongly exposes BX if any minimizing sequence of w∗ in BX has a converging subsequence. Proposition 19 ([11, Cor. 3.6]) If dC is Fréchet subdiﬀerentiable at w ∈ X\C and if some w∗ ∈ ∂ − dC (w) strongly exposes BX , then P (C, w) is nonempty and P (C, w) = {w∗ − S∗ (w∗ )}, where S∗ (w∗ ) is the point in SX at which w∗ attains its maximum. In particular, if the dual norm on X ∗ is Fréchet diﬀerentiable at some w∗ ∈ ∂ − dC (w), then P (C, w) is nonempty. This result has been given in [9] for a Hilbert space, in [4] in a reﬂexive Banach space satisfying the Kadec–Klee property, and in [18] in a Banach space with a uniformly Gâteaux diﬀerentiable norm under a diﬀerentiability condition on dC ; see also [25]. Example 5.4 in [18] shows that Fréchet diﬀerentiability of dC is not enough to ensure the existence of nearest points. Let us now recall some criteria ensuring Gâteaux or Fréchet diﬀerentiability of dC . Proposition 20 ([18], see also [25]) If, for some w ∈ X\C, any minimizing sequence of · − w in C converges to some x ∈ X and if the norm of X is Gâteaux (resp. Fréchet) diﬀerentiable at w − x, then dC is Gâteaux (resp. Fréchet) diﬀerentiable at w. The following criterion is a slight variant of [17, Cor. 2.6]; it stems from an inspection of the proof of [17, Thm. 2.4]. Proposition 21 If, for some w ∈ X\C, u ∈ SX one has dC (x, u) = 1 and (−dC ) (x, −u) = 1, in particular, if dC has a directional derivative at x in the direction u with value 1, and if the norm of X is Fréchet diﬀerentiable at u with derivative u∗ then dC is Fréchet diﬀerentiable at x with derivative u∗ . Let us reformulate a notion introduced in [25] and implicitely used in [9]. Definition 22 The halo H(C) of a nonempty closed subset C is the set of w ∈ X \ C such that there exist x ∈ P (C, w) and s ∈ (0, +∞) with x ∈ P (C, w + s (w − x)). Proposition 23 If C is a nonempty closed convex subset of X, then H(C) is the set of points of X \ C having a best approximation in C. Proof The result follows from the fact that, for any w ∈ X \ C, x ∈ P (C, w) and any r > 0, x is a best approximation of w + r (w − x) in C ([43]). It is easy to show that, for any closed subset C of X, when w0 ∈ X\C is such that P (C, w0 ) is nonempty, then, for any x ∈ P (C, w0 ) and any t ∈ (0, 1), one has w := x + t(w0 − x) ∈ H(C). Moreover, in such a case, dC has a derivative at w in the direction u := w − x with value 1, so that Proposition 21 yields the Fréchet case of the next statement. Proposition 24 ([25, Prop. 1.5]) Let C be a nonempty closed subset of X. Assume that w ∈ H(C) and the norm of X is Gâteaux (resp. Fréchet) diﬀerentiable at w −x, where x ∈ P (C, w). Then dC is Gâteaux (resp. Fréchet) diﬀerentiable at w with derivative dC (w) = S(w − x). Thus, when the norm of X is Gâteaux (resp. Fréchet) diﬀerentiable oﬀ 0, and C is a sun in the sense that H(C) = X \ C (a terminology introduced in [3]), the distance function is Gâteaux (resp. Fréchet) diﬀerentiable on X \ C. Corollary 25 Let C be a nonempty sun of X. If X ∗ is strictly convex, then C is convex. Distance Functions 515 Proof The result follows from the fact ([3, Thm. 18]) that a nonempty closed subset C of X is convex whenever dC is Gâteauxdiﬀerentiable on X \ C and dC (w) = 1 for any w ∈ X \ C. 5 Enlargements and Distance Functions In this section, we exhibit assumptions on the closed subset C so that inclusion (6) (or some analogue) turns out to be an equality. Moreover, we consider the regularizations of C (as in [5], [7], [9], [16]) obtained by enlargements. Definition 26 For r ≥ 0, we set Cr := {u ∈ X : dC (u) ≤ r} and Cr := C + rBX . The set Cr is called the renlargement of C. Basic properties for Cr and Cr follow from this deﬁnition. Note that Cr = Cr iﬀ for each w ∈ (dC )−1 (r) one has P (C, w) = ∅, an assumption we do not want to adopt from the outset, even if it is satisﬁed when C is boundedly compact, in particular if X is ﬁnite dimensionaL. Lemma 27 For each r ≥ 0, the set Cr is the closure cl(Cr ) of Cr . If, moreover, X is reﬂexive and C is weakly closed, in particular if dim X < +∞, then Cr = Cr . Proof Let r ≥ 0. Obviously Cr is closed and Cr := C + rBX ⊂ Cr , so that cl(Cr ) ⊂ Cr . Conversely, let w ∈ Cr . For each ε > 0, there exists xε ∈ C such that xε − w ≤ dC (w) + ε ≤ r (w − xε ). Then yε ∈ C + rBX = Cr and r + ε. Let us set yε := xε + r+ε r ε yε − w = − 1 xε − w ≤ (r + ε) = ε. r+ε r+ε Thus w ∈ cl (C + rBX ) = Cr . The last assertion is obvious. Corollary 28 The set Cr is locally quasistarshaped at w ∈ Cr if, and only if, the set Cr is locally quasistarshaped at w. Proof Since Cr is the closure of Cr one has T (Cr , w) = T (Cr , w) and, for any open subset V containing 0, one has (Cr − w) ∩ V = cl(Cr − w) ∩ V ⊂ cl((Cr − w) ∩ V ), so that (Cr − w) ∩ V ⊂ T (Cr , w) ⇔ (Cr − w) ∩ V ⊂ T (Cr , w) . The next result being crucial for the following, we give a proof for the reader’s convenience, although its ﬁrst part is probably known (and its second part can be deduced from part (a) and the obvious semigroup property of r → Cr ). Here, for a real number t, we set t+ := max(t, 0). Lemma 29 Let C be a nonempty closed subset of a Banach space X. + (a) For each r ≥ 0, one has dCr = (dC − r) = dCr ; (b) For any r, s ≥ 0, one has (Cr )s = Cr+s . Proof (a) Clearly, dCr = dCr by Lemma 27. Let w ∈ X. For any s > dCr (w), there exists z ∈ Cr such that w − z < s. It follows that dC (w) − r ≤ w − z + dC (z) − r < w − z < s, + and we get (dC − r) ≤ dCr . On the other hand, for any t > (dC (w) − r)+ , we can ﬁnd x ∈ C such that x − w < r + t and setting y := x + r(r + t)−1 (w − x) we have y ∈ Cr and + y − w ≤ t(r + t)−1 x − w ≤ t, so that dCr (w) ≤ (dC (w) − r) . + (b) Given x ∈ (Cr )s we have dCr (x) ≤ s, hence dC (x) − r ≤ (dC (x) − r) ≤ s, so that dC (x) ≤ r + s and x ∈ Cr+s . Conversely, given x ∈ Cr+s , we have dC (x) − r ≤ s, hence dCr (x) = (dC (x) − r)+ ≤ s, so that x ∈ (Cr )s . The next inclusions are easy consequences of Lemma 29. They show a regularizing eﬀect of enlargements. Proposition 30 Let w ∈ X \ C and let r = dC (w). For any x ∈ P (C, w) one has T (C, x) ⊂ T (Cr , w) and N (Cr , w) ⊂ N (C, x). Moreover, N − (Cr , w) ⊂ N − (C, x). Penot J. P. and Ratsimahalo R. 516 Proof u ∈ T (C, x). There exist sequences (tn ) → 0+ , (un ) → u such that x + tn un ∈ C. As r = dC (w) = w − x and dC (w + tn un ) − r ≤ dC (x + tn un ) = 0, Lemma 29 implies that dCr (w + tn un ) = (dC (w + tn un ) − r)+ = 0, so that u ∈ T (Cr , w) . The inclusion N (Cr , w) ⊂ N (C, x) follows by polarity. Given w∗ ∈ N − (Cr , w) , for any ε > 0 one can ﬁnd δε > 0 such that w∗ , u − w ≤ ε u − w , ∀ u ∈ Cr ∩ B(w, δε ). For any v ∈ C ∩ B(x, δε ), we have u := v + w − x ∈ Cr ∩ B(w, δε ), so that w∗ , v − x = w∗ , u − w ≤ ε u − w = ε v − x . Thus, w∗ ∈ N − (C, x). We are now able to get more information about the Fréchet subdiﬀerential of dC at a point w ∈ X \ C. We also consider the (strong) limiting subdiﬀerential ∂ − dC (w) of dC at w and the (strong) limiting normal cone to C respectively given by ∂ − dC (w) := {w∗ ∈ X ∗ : ∃(wn ) → w, ∃(wn∗ ) → w∗ , ∀ n ∈ N, wn∗ ∈ ∂ − dC (wn )}, N − (C, w) := {w∗ ∈ X ∗ : ∃(wn ) → w, ∃(wn∗ ) → w∗ , ∀ n ∈ N, wn ∈ C, wn∗ ∈ N − (C, wn )}. Let w ∈ X \ C and let r = dC (w). Then ∂ − dC (w) ⊂ N − (Cr , w) ∩ SX ∗ , (8) ∂dC (w) ⊂ N (Cr , w) ∩ BX ∗ . (9) If, moreover, P (C, w) = ∅, then, for each x ∈ P (C, w) such that · is Fréchet diﬀerentiable at w − x, one has (10) ∂ − dC (w) = N − (Cr , w) ∩ S(w − x). Theorem 31 If X is an Asplund space one has ∂ − dC (w) ⊂ N − (Cr , w) ∩ SX ∗ ⊂ N − (Cr , w) ∩ SX ∗ = ∂ − dC (w). Proof For w ∈ X \C we already know that ∂ − dC (w) ⊂ SX ∗ and ∂dC (w) ⊂ BX ∗ by Lemma 12. Now, by an easy computation and by Lemma 29 and Proposition 8 one has ∂ − dC (w) = ∂ − (dC − r)(w) ⊂ ∂ − (dC − r)+ (w) = ∂ − dCr (w) = BX ∗ ∩ N − (Cr , w) . Let w∗ ∈ ∂dC (w). Since dCr (w) = 0 and dC (w) = r, Lemma 29 implies that for any u ∈ X one has dCr (w, u) = (dC (w, u))+ . It follows that for any u ∈ T (Cr , w), one has dC (w, u) ≤ 0. Hence w∗ , u ≤ 0 for any u ∈ T (Cr , w). Thus w∗ ∈ N (Cr , w). When x ∈ P (C, w) we have ∂ − dC (w) ⊂ ∂dC (w) ⊂ S(w − x) by relation (6) and thus inclusion (8) can be reﬁned by replacing SX ∗ by S(w − x) = S(u) with u := r −1 (w − x). In order to prove the reverse inclusion when · is Fréchet diﬀerentiable at w − x, let us pick w∗ in N − (Cr , w) ∩ S(u). Given ε > 0, let δ > 0 be such that 1 for z ≤ δ. u − z − u − w∗ , z ≤ ε z 2 By Proposition 8, since w∗ ∈ S(u) = J(u), hence u ∈ J −1 (N − (Cr , w)), we can ﬁnd τ > 0 such that, for t ∈ [0, τ ], we have dCr (w + tu) − t ≤ 12 εt. Then, dC (w + tu) − r − t ≤ 12 εδt. Since dC is Lipschitzian with rate one, we have, for v ∈ τ δBX , t := δ −1 v ∈ [0, τ ], z := t−1 v ∈ δBX dC (w + v) − dC (w) = dC (w + v) − dC (w + tu) + dC (w + tu) − dC (w) 1 1 ≥ − tu − v + t − εδt = −t u − t−1 v − u − εδt 2 2 1 1 ∗ −1 −1 ∗ ≥ −tw , −t v − tε t v − ε v = w , v − ε v . 2 2 Thus w∗ ∈ ∂ − dC (w). Distance Functions 517 Now let us suppose X is an Asplund space and let w∗ ∈ N − (Cr , w)∩SX ∗ . By [44, Thm. 4.1] one can ﬁnd some sequences (wn ) → w in X, (wn∗ ) in X ∗ , (rn ) in R+ such that (rn wn∗ ) → w∗ and wn∗ ∈ ∂ − dC (wn ) for any n ∈ N. Then, we have (rn ) = (rn wn∗ ) → w∗ = 1, hence (wn∗ ) → w∗ and w∗ ∈ ∂ − dC (w). Now, if w∗ ∈ N − (Cr , w) ∩ SX ∗ we can ﬁnd sequences (wn ) in Cr , (wn∗ ) → −1 w∗ such that wn∗ ∈ N − (Cr , wn ) for each n ∈ N. Then (vn∗ ) := (wn∗ wn∗ ) → w∗ and vn∗ ∈ ∗ − − ∂ dC (wn ) by what precedes. It follows that w ∈ ∂ dC (w). The reverse inclusion ∂ − dC (w) ⊂ N − (Cr , w) ∩ SX ∗ follows from the deﬁnitions and the inclusion ∂ − dC (w) ⊂ N − (Cr , w) ∩ SX ∗ . The following consequence has been obtained in [9, Thm. 3.4] in a Hilbert space by the way of a delicate use of the scalar product. Here we make assumptions which are valid in more general spaces. Corollary 32 Let X be a reﬂexive Banach space with a Fréchet diﬀerentiable norm. Then, for any w ∈ X \ C one has, with r := dC (w), ∂ − dC (w) = N − (Cr , w) ∩ SX ∗ . Proof Let w ∈ X \ C, r := dC (w), w∗ ∈ N − (Cr , w) ∩ SX ∗ . Since X is reﬂexive, we can ﬁnd some u ∈ X such that w∗ , u = u = 1. Since the norm is diﬀerentiable, we have w∗ = S(u). The preceding proof (which does not use the fact that w − ru ∈ C) shows that w∗ ∈ ∂ − dC (w). Under a geometric assumption on C instead of assumptions on the geometry of X, the inclusions (8) and (9) also turn out to be equalities. Corollary 33 If Cr is pseudoconvex at w ∈ X \ C, with r = dC (w), then N − (Cr , w) = N (Cr , w) and ∂ − dCr (w) = N (Cr , w) ∩ SX ∗ = N − (Cr , w) ∩ SX ∗ = ∂dC (w) ∩ SX ∗ . (11) If, moreover, P (C, w) = ∅ then ∂dC (w) = x∈P (C,w) N (Cr , w) ∩ S(w − x) = ∂ − dC (w). (12) Proof The ﬁrst equality is a direct application of Propositions 3 and 9 obtained by replacing C with Cr . Let w∗ ∈ N (Cr , w) ∩ SX ∗ . Suppose that Cr is pseudoconvex at w. Then, for any u ∈ BX , x ∈ C, one has x + ru − w ∈ T (Cr , w) , thus w∗ , x − w + rw∗ , u = w∗ , x + ru − w ≤ 0. Taking the supremum over u ∈ BX and using the fact that w∗ = 1 and r = dC (w), one obtains that, for any z ∈ X, w∗ , z − x + w∗ , x − w + dC (w) ≤ w∗ , z − x ≤ z − x . Gathering terms on the lefthand side and passing to the inﬁmum over x ∈ C, it follows that w∗ , z − w + dC (w) ≤ inf x∈C z − x = dC (z) . (13) ∗ ∗ − Thus w ∈ ∂dC (w) in the sense of convex analysis, hence w ∈ ∂ dC (w). Then, by this inclusion, (9) and (8), ∂ − dC (w) ⊂ ∂dC (w) ∩ SX ∗ ⊂ N (Cr , w) ∩ SX ∗ = N − (Cr , w) ∩ SX ∗ ⊂ ∂ − dC (w). The last assertion follows from the inclusion S(w−x) ⊂ SX ∗ for x ∈ P (C, w), and from relations (6), (9), (11). When X is a Hilbert space, and the Hadamard subdiﬀerential is replaced by the proximal subdiﬀerential, relation (12) is proved in [9] without the assumption that C is pseudoconvex, but with a crucial use of the inner product. 6 The Convex Case As noted above, under convexity assumptions, the preceding results take more striking forms. The following result follows from Corollary 33 and the fact that when C is convex, Cr is convex, and hence pseudoconvex at each point. Moreover, the formula ∂dC (w) = BX ∗ ∩ N (C, w), if w ∈ C is well known in such a case. Corollary 34 Suppose C is convex. Then, for w ∈ X \ C, r = dC (w), one has ∂dC (w) = N (Cr , w) ∩ SX ∗ = N − (Cr , w) ∩ SX ∗ = ∂dC (w) ∩ SX ∗ . (14) Penot J. P. and Ratsimahalo R. 518 If P (C, w) = ∅ then ∂dC (w) = x∈P (C,w) S(w − x) ∩ N (Cr , w) . Relation (14) is similar to the one stated in [5] except that the enlargement used in [5] (namely Cr ) is not necessarily closed in an inﬁnitedimensional space, whereas our enlargement is always closed. Speciﬁc diﬀerentiability results for dC hold under the assumption that C is convex. The ﬁrst one follows from Propositions 23 and 24. Corollary 35 Let C be a nonempty closed convex subset of X and let w ∈ X\C. Assume there exists x ∈ P (C, w). If the norm of X is Gâteaux (resp. Fréchet) diﬀerentiable at w − x, then dC is Gâteaux (resp. Fréchet) diﬀerentiable at w with {dC (w)} = {S(w − x)}. In particular, if X is reﬂexive and if its norm is Gâteaux diﬀerentiable oﬀ 0, then dC is Gâteaux diﬀerentiable on X \ C and, for any w ∈ X \ C, one has {dC (w)} = x∈P (C,w) {S(w − x)} . Another result about Fréchet diﬀerentiability can be obtained by using a classical notion and a quantitative tool connected with it. The norm of a Banach space X is said to be locally uniformly convex at x if, for any sequence (xn ) such that (xn ) converges to x and (xn + x) converges to 2 x, the sequence (xn − x) converges to 0. A modulus of (local) uniform convexity at x ∈ SX is a modulus μ (i.e. a nondecreasing function on R+ null at 0 and continuous at 0) such that, for w ∈ SX , one has 1 w − x ≤ μ 1 − w + x . 2 A Banach space X is said to be locally uniformly convex if it is locally uniformly convex at each point (see [37], [45], [46]). The qualitative part of the following statement is known (see for instance [37], [38]). However the result we present is quantitative. Proposition 36 Let C be a nonempty closed convex subset of X and let w ∈ X \ C, r := dC (w). If there exists w∗ ∈ ∂dC (w), at which the norm of the dual X ∗ is locally uniformly convex, then dC is Fréchet diﬀerentiable at w with derivative dC (w) = w∗ . Moreover, if μ is a modulus of uniform convexity of the dual norm at w∗ , one has, for v ∈ X \ C, ∂dC (v) ⊂ {dC (w)} + μ r −1 v − w BX ∗ . Proof Let w,v ∈ X \ C and let (w∗ , v ∗ ) ∈ ∂dC (w) × ∂dC (v). Let (xn ) be a sequence of C such that (xn − w)n converge to dC (w). By the deﬁnition of ∂dC , one has w∗ , w − xn ≥ dC (w), v ∗ , w − xn ≥ dC (v) + v ∗ , w − v. Adding side by side and dividing by rn := w − xn we get, with un := rn−1 (w − xn ), 2 ≥ w∗ + v ∗ X ∗ ≥ w∗ + v ∗ , un ≥ rn−1 (dC (w) + dC (v) + v ∗ , w − v) . ∗ Since v X ∗ = 1 and dC (v) ≥ dC (w) − w − v , taking the limit as n → +∞, we get w∗ + v ∗ X ∗ ≥ 2 − 2r −1 w − v. The local uniform convexity of the dual norm at w∗ implies that w∗ − v ∗ ≤ μ(r −1 w − v). 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