Continuous selections of Lipschitz extensions in metric spaces

arXiv:1502.06842v1 [math.MG] 25 Nov 2014 Continuous Selections of Lipschitz Extensions in Metric Spaces Rafa Esp´ınolaa, Adriana Nicolaeb,c aDepartamento de An´alisis Matem´atico - IMUS, Universidad de Sevilla, Apdo. de Correos 1160, 41080 Sevilla, Spain bDepartment of Mathematics, Babe¸s-Bolyai University, Kog˘alniceanu 1, 400084 Cluj-Napoca, Romania cSimion Stoilow Institute of Mathematics of the Romanian Academy, Research group of the project PD-3-0152, P.O. Box 1-764, RO-014700 Bucharest, Romania E-mail addresses: [email protected] (R. Esp´ınola), [email protected].ro (A. Nicolae) Abstract This paper deals with the study of parameter dependence of extensions of Lipschitz mappings from the point of view of continuity. We show that if assuming appropriate curvature bounds for the spaces, the multivalued extension operators that assign to every nonexpansive (resp. Lipschitz) mapping all its nonexpansive extensions (resp. Lipschitz extensions with the same Lipschitz constant) are lower semi-continuous and admit continuous selections. Moreover, we prove that Lipschitz mappings can be extended continuously even when imposing the condition that the image of the extension belongs to the closure of the convex hull of the image of the original mapping. When the target space is hyperconvex one can obtain in fact nonexpansivity. Keywords: Lipschitz mapping, extension operator, continuous selection, geodesic space of bounded curvature, hyperconvexity 1 Introduction The Kirszbraun theorem [12] is a fundamental result in the theory of Lipschitz extensions and states that for any Lipschitz function f:A⊆Rn→Rmthere exists a Lipschitz extension f′:Rn→Rmwith the same Lipschitz constant. The result for arbitrary Hilbert spaces goes back to Valentine [22]. Aronszajn and Panitchpakdi [3] introduced the concept of hyperconvexity, which is closely related to this problem since a metric space Yis hyperconvex if and only if given any subspace Aof any metric space X, every nonexpansive mapping f:A→Yadmits a nonexpansive extension to X. The first result that extends Kirszbraun’s theorem to the metric setting by imposing curvature bounds in the sense of Alexandrov was given by Lang and Schroeder in [17] (see also [16, 22] for previous particular results). The same problem was later approached by Alexander, Kapovitch and Petrunin in [2] where a different proof method is considered. All the aforementioned results guarantee the existence of an extension for the original mapping. However, this extension is not necessary unique and no information is given on the parameter dependence of the extensions. Kopeck´a studied the process of assigning extensions to mappings from the point of view of continuity providing positive answers first in Euclidean [13] and then in Hilbert spaces [14]. Namely, the multivalued extension mappings that assign to every nonexpansive (resp. Lipschitz) mapping all its nonexpansive extensions (resp. Lipschitz extensions with the same Lipschitz constant) are proved to be lower semi-continuous using Kirszbraun’s theorem and a homotopy argument. Applying Michael’s selection theorem one obtains continuous selections of these multivalued extension operators. Kopeck´a and Reich further generalized these results in [15], obtaining a continuous singlevalued extension operator with the additional condition that the image of the extension belongs to the closed convex hull of the image of the original mapping. A natural question is to study this problem in geodesic metric spaces with curvature bounds in the sense of Alexandrov since in this context a generalized version of Kirszbraun’s theorem holds. Here we show that one can indeed prove counterparts of such continuity results in this setting. In Section 3 we show that assuming appropriate curvature bounds for the spaces, the multivalued extension mappings are lower 1 semi-continuous and admit continuous selections. Moreover, we prove in Section 4 that Lipschitz mappings can be extended continuously even when imposing the above mentioned convexity condition on the image of the extension. Section 5 deals with the case where the target space is hyperconvex and shows that in this situation one can obtain in fact nonexpansivity. 2 Preliminaries Let (X, d) be a metric space. A geodesic path from xto yis a mapping c: [0, l]⊆R→Xsuch that c(0) = x, c(l) = yand d(c(t), c(t′)) = |t−t′|for every t, t′∈[0, l]. The image c([0, l]) of cforms a geodesic segment which joins xand y. Note that a geodesic segment from xto yis not necessarily unique. (X, d) is a geodesic space if every two points in Xcan be joined by a geodesic path. A point z∈Xbelongs to a geodesic segment joining xand yif and only if there exists t∈[0,1] such that d(z, x) = td(x, y) and d(z, y) = (1 −t)d(x, y), and we will write z= (1 −t)x+ty for simplicity. For more details on geodesic metric spaces the reader may check [4]. A geodesic space (X, d) is Busemann convex if given any pair of geodesic paths c1: [0, l1]→Xand c2: [0, l2]→Xwith c1(0) = c2(0) one has d(c1(tl1), c2(tl2)) ≤td(c1(l1), c2(l2)),for every t∈[0,1]. A subset Cof Xis convex if any geodesic segment that joins every two points of Cis contained in C. Let G1(C) denote the union of all geodesics segments with endpoints in C. Note that Cis convex if and only if G1(C) = C. Recursively, for n≥2 we set Gn(C) = G1(Gn−1(C)). The convex hull of Cis co(C) = [ n∈N Gn(C). By co(C) we denote the closure of the convex hull. It is easy to see that in a Busemann convex geodesic space, the closure of the convex hull is convex and hence it is the smallest closed convex set containing C. For κ∈Rlet M2 κdenote the complete, simply connected model surface of constant curvature κ. In the sequel we assume that κ≤0. Ageodesic triangle ∆ = ∆(x1, x2, x3) consists of three points x1, x2and x3in Xand three geodesic segments corresponding to each pair of points. A κ-comparison triangle for ∆ is a triangle ¯ ∆ = ∆(¯x1,¯x2,¯x3) in M2 κsuch that d(xi, xj) = dM2 κ(¯xi,¯xj) for i, j ∈ {1,2,3}. For κfixed, κ-comparison triangles of geodesic triangles always exist and are unique up to isometry. A geodesic triangle ∆ satisfies the CAT(κ) (resp. reversed CAT(κ)) inequality if for every κ-comparison triangle ¯ ∆ of ∆ and for every x, y ∈∆ we have d(x, y)≤dM2 κ(¯x, ¯y) (resp. d(x, y)≥dM2 κ(¯x, ¯y)), where ¯x, ¯y∈¯ ∆ are the corresponding points of xand y, i.e., if x= (1 −t)xi+txjthen ¯x= (1 −t)¯xi+t¯xj. ACAT(κ)space (also known as a space of curvature bounded above by κin the sense of Alexandrov) is a geodesic space for which every geodesic triangle satisfies the CAT(κ) inequality. Any CAT(0) space (and so any CAT(κ) space) is Busemann convex. A geodesic metric space is said to have curvature bounded below by κin the sense of Alexandrov (denoted by CBB(κ)) if every geodesic triangle satisfies the reversed CAT(κ) inequality. If Xis a CBB(κ) space, then the direct product X×M2 κis a CBB(κ) space with the metric d((x, a),(y, b))2=dX(x, y)2+dM2 κ(a, b)2.(1) Other properties of spaces with curvature bounded above or below and equivalent definitions can be found in [4, 5]. Let (X, d) be a metric space. Taking x∈Xand r > 0 we denote the closed ball centered at xwith radius rby B(z, r).Given Ca nonempty subset of X, the distance of a point x∈Xto Cis dist(x, C) = inf{d(x, c) : c∈C}.If Band Care nonempty subsets of X, one defines the Pompeiu-Hausdorff distance as H(B, C) = max sup b∈B dist(b, C),sup c∈C dist(c, B). 2 The metric projection PConto Cis the mapping PC(x) = {c∈C:d(x, c) = dist(x, C)},for every x∈X. In any CAT(0) space the metric projection onto a convex and complete subset is a singlevalued and nonexpansive (that is, 1-Lipschitz) mapping. A metric space Xis hyperconvex if TαB(xα, rα)6=∅for every collection of points {xα}in Xand positive numbers {rα}such that d(xα, xβ)≤rα+rβfor any α, β. A subset Eof a metric space Xis called externally hyperconvex (with respect to X) if given any family {xα}of points in Xand any family {rα}of real numbers satisfying d(xα, xβ)≤rα+rβand dist(xα, E)≤rα, it follows that TαB(xα, rα)∩E6=∅. For a more detailed discussion on hyperconvex metric spaces, see [6]. Let (X, dX), (Y, dY) be metric spaces, A⊆Xnonempty and consider C(A, Y ) the family of bounded and continuous mappings from Ato Y. For each f, g ∈C(A, Y ), let d∞(f, g) = supx∈AdY(f(x), g(x)). Endowed with the supremum distance d∞,C(A, Y ) is a metric space which is complete if Yis complete. We consider two subsets of C(A, Y ): L(A, Y ) which includes all bounded Lipschitz mappings from Ato Yand is not necessarily a closed subset of C(A, Y ) and N(A, Y ) which stands for the family of all bounded nonexpansive mappings defined from Ato Yand which is closed in C(A, Y ). For f∈L(A, Y ) we denote the smallest Lipschitz constant of fon B⊆Aby Lip(f, B). More precisely, Lip(f, B) = sup dY(f(x), f(y)) dX(x, y):x, y ∈B, x 6=y. For a set C, we denote by P(C) the family of all its subsets. We consider two multivalued extension mappings: •Φ : N(A, Y )→P(N(X, Y )) which assigns to each nonexpansive mapping f∈N(A, Y ) all its nonexpansive extensions f′∈N(X, Y ). Note that in this case it may happen that Lip(f, A)< Lip(f′, X)≤1. •Ψ : L(A, Y )→P(L(X, Y )) which assigns to each Lipschitz mapping f∈L(A, Y ) all its Lipschitz extensions f′∈L(X, Y ) with Lip(f, A) = Lip(f′, X). Recall that having two topological spaces Xand Y, a multivalued mapping Γ : X→P(Y) is lower semi-continuous if for every open V⊆Y, the set {x∈X: Γ(x)∩V6=∅} is open in X. If Xand Yare metric spaces, Γ is nonexpansive if H(Γ(x),Γ(y)) ≤dX(x, y) for every x, y ∈X. The classical Kirszbraun theorem was extended to geodesic metric spaces with lower and upper curvature bounds by Lang and Schroeder in [17]. Later, Alexander, Kapovitch and Petrunin considered a different approach of the proof in [2]. Theorem 2.1 (Lang, Schroeder [17]).Let κ≤0,Xa CBB(κ)space and Ya complete CAT(κ)space. Suppose A⊆Xis nonempty and f:A→Yis nonexpansive. Then there exists a nonexpansive extension f′:X→Yof f. Although the result can be also stated when κ > 0 with an appropriate boundedness condition on the set f(A), here we are only concerned with the case κ≤0. For κ= 0 the result can be generalized to any arbitrary Lipschitz constant by scaling the metric on either Xor Yand so we may consider both mappings Φ and Ψ. When κ < 0, the same argument can be applied for Lipschitz constants greater than 1. However, for Lipschitz constants strictly less than 1, we cannot expect the result to hold true. Suppose one could extend all mappings f:A⊆H2→H2with Lip(f, A)<1 while keeping the same Lipschitz constant. Taking κ∈(−1,0), this implies that we can extend all nonexpansive mappings defined on A⊆H2with values in M2 κto nonexpansive mappings on H2.But this means that M2 κ is a CAT(−1) space (see Proposition 6.2 in [17]), a contradiction. Since in this work we rely on Theorem 2.1 in order to obtain our continuity results, for the case κ < 0 we will only study the mapping Φ. However, if the target space is an R-tree, then it was proved in [17] that we not only can extend mappings with arbitrary Lipschitz constant, but we can also drop the curvature assumption on the source space. 3 Theorem 2.2 (Lang, Schroeder [17]).Let Xbe a metric space and Ya complete R-tree. Suppose A⊆X is nonempty and f:A→Yis a Lipschitz mapping. Then there exists a Lipschitz extension f′:X→Yof fwith Lip(f′, X) = Lip(f, A). Theorem 2.2 is a consequence of the following extension theorem proved for hyperconvex metric spaces by Aronszajn and Panitchpakdi in [3], where it is actually shown that this property characterizes hyperconvexity. Note that any complete R-tree is a hyperconvex metric space (see [11]). Theorem 2.3 (Aronszajn, Panitchpakdi [3]).Let Xbe a metric space and Ya hyperconvex metric space. Suppose A⊆Xis nonempty and f:A→Yis a Lipschitz mapping. Then there exists a Lipschitz extension f′:X→Yof fwith Lip(f′, X) = Lip(f, A). 3 Lower semicontinuity of the multivalued extension mappings and continuous selections We begin this section by showing that, when considering appropriate curvature bounds on Xand Y, both mappings Φ and Ψ are lower semi-continuous which is an immediate consequence of Lemmas 3.1 and 3.2, respectively. The proof strategy follows the one used for Hilbert spaces in [14]. Lemma 3.1. Let κ≤0,Xa CBB(κ)space, Ya complete CAT(κ)space and A⊆Xnonempty. Let f∈ N(X, Y ). Then for every ε > 0there exists δ > 0such that every g∈N(A, Y )with supa∈AdY(f(a), g(a)) < δadmits an extension g′∈N(X, Y )such that d∞(f, g′)≤ε. Proof. Since fis a bounded mapping there exists z∈Yand M≥1 such that supx∈XdY(z, f(x)) ≤M. Let ε∈(0,1) and take δ=ε2/(8M). Suppose g∈N(A, Y ) with supa∈AdY(f(a), g(a)) < δ. Let κ= 0. Define the mapping h:X× {(0,0)}∪A×{(0, ε)} → Yby: for x∈X,h(x, (0,0)) = f(x) and for a∈A,h(a, (0, ε)) = g(a). Recalling (1), for x∈Xand a∈A, dY(h(x, (0,0)) , h (a, (0, ε)))2=dY(f(x), g(a))2 ≤(dY(f(x), f(a)) + dY(f(a), g(a)))2 ≤dX(x, a)2+δ2+ 4δM < dX(x, a)2+ε2=d((x, (0,0)),(a, (0, ε)))2. This shows that his nonexpansive since both fand gare nonexpansive on Xand A, respectively. Since X×R2is a CBB(0) space, using Theorem 2.1 we can extend hto a nonexpansive mapping h′:X×R2→Y. Define g′:X→Yby g′(x) = h′(x, (0, ε)). Clearly, g′is nonexpansive and coincides with gon A. Moreover, for each x∈X, dY(f(x), g′(x)) = dY(h′(x, (0,0)) , h′(x, (0, ε))) ≤d((x, (0,0)),(x, (0, ε))) = ε. This also shows that g′is bounded. When κ < 0, we apply the same argument to the nonexpansive mapping h:X×{(0,0,1)}∪A×0,sinh √−κε,cosh √−κε→Y defined as: for x∈X,h(x, (0,0,1)) = f(x) and for a∈A,ha, 0,sinh √−κε,cosh √−κε=g(a) which can be extended to a nonexpansive mapping h′:X×M2 κ→Y(recall that X×M2 κis a CBB(κ) space). Lemma 3.2. Let Xbe a CBB(0) space, Ya complete CAT(0) space and A⊆Xnonempty. Let f∈L(X, Y ) with Lip(f, A) = Lip(f, X). Then for every ε > 0there exists δ > 0such that every g∈L(A, Y )for which supa∈AdY(f(a), g(a)) < δ admits an extension g′∈L(X, Y )with Lip(g, A) = Lip(g′, X)and d∞(f, g′)≤ε. 4 Proof. Let ε∈(0,1). Suppose first that fis constant and equal to some y∈Y. Let δ=ε. Then having any extension g1of gto Xwith Lip(g, A) = Lip(g1, X), we can take g′:X→Y,g′(x) = PB(y,ε)◦g1. Assume now fis not constant. Let z∈Yand M > 0 such that sup x∈X dY(z, f(x)) ≤M. Let s∈(0,1) for which 1−s s2<ε2 32M(4M+ 1). Since Lip(f, A) = Lip(f, X)>0, there exist x0, y0∈Asuch that dY(f(x0), f(y0)) > sLip(f, X)dX(x0, y0). Take δ= min dY(f(x0), f(y0)) −sLip(f, X)dX(x0, y0) 2,ε2s2 32(4M+ 1). Let g∈L(A, Y ) with supa∈AdY(f(a), g(a)) < δ. Suppose first Lip(g, A)≤2Lip(f, X). Then, dY(g(x0), g(y0)) ≥dY(f(x0), f(y0)) −dY(f(x0), g(x0)) −dY(f(y0), g(y0)) > dY(f(x0), f(y0)) −2δ≥sLip(f, X)dX(x0, y0), from where Lip(g, A)≥sLip(f, X). Let η=ε/(4Lip(f, X)) and h:X× {(0,0)} ∪ A× {(0, η)} → Ybe defined by: for x∈X,h(x, (0,0)) = (1 −s)z+sf(x) and for a∈A,h(a, (0, η)) = g(a). Thus, for x∈X and a∈Awe have that dY(h(x, (0,0)) , h (a, (0, η)))2=dY((1 −s)z+sf(x), g(a))2 ≤(dY((1 −s)z+sf(x), f(a)) + dY(f(a), g(a)))2 ≤((1 −s)M+sdY(f(x), f(a)) + δ)2 ≤(δ+ (1 −s)M)2+s2Lip(f, X)2dX(x, a)2+ 4sM (δ+ (1 −s)M) < s2Lip(f, X)2dX(x, a)2+(δ+ (1 −s)M)(4M+ 1) s2Lip(f, X)2 since (δ+ (1 −s)M)2< δ + (1 −s)Mand s < 1 < s2Lip(f, X)2dX(x, a)2+η2 since δ+ (1 −s)M < ε2s2/(16(4M+ 1)) ≤Lip(g, A)2d((x, (0,0)),(a, (0, η)))2. To complete the argument that his Lipschitz with smallest Lipschitz constant Lip(g, A) one uses Busemann convexity in Yalong with the fact that the mappings fand gare Lipschitz and Lip(g, A)≥sLip(f, X). Since X×R2is a CBB(0) space, by Theorem 2.1 we can extend hto a Lipschitz mapping h′:X×R2→Y with Lip(h′, X ×R2) = Lip(g, A). Define g′:X→Yby g′(x) = h′(x, (0, η)). Clearly, g′extends gand Lip(g′, X) = Lip(g, A). Moreover, for every x∈X, dY(g′(x), f(x)) ≤dY(g′(x),(1 −s)z+sf(x)) + dY((1 −s)z+sf(x), f(x)) ≤dY(h′(x, (0, η)) , h′(x, (0,0))) + (1 −s)M < Lip(g, A)η+ε/2 ≤2Lip(f, X)ε 4Lip(f, X)+ε 2=ε. If Lip(g, A)>2Lip(f, X), consider the set ˜ A=x∈X: dist(x, A)≥2δ Lip(g, A) 5 and define the mapping ˜g:A∪˜ A→Yby: for a∈A, ˜g(a) = g(a) and for x∈˜ A, ˜g(x) = f(x). To see that Lip(g, A) = Lip(˜g, A ∪˜ A) it suffices to verify that for any a∈Aand x∈˜ A, dY(˜g(x),˜g(a)) = dY(f(x), g(a)) ≤dY(f(x), f(a)) + dY(f(a), g(a)) <Lip(g, A) 2dX(x, a) + δ≤Lip(g, A) 2dX(x, a) + Lip(g, A) 2dist(x, A) ≤Lip(g, A)dX(x, a). Take g′to be any extension of ˜gfor which Lip(g, A) = Lip(g′, X). For x∈˜ A,f(x) = g′(x). If x /∈˜ A, there exists a∈Asuch that dX(x, a)<2δ/Lip(g, A). Thus, dY(f(x), g′(x)) ≤dY(f(x), f(a)) + dY(f(a), g′(a)) + dY(g′(a), g′(x)) <Lip(g, A) 2 2δ Lip(g, A)+δ+ Lip(g, A)2δ Lip(g, A)= 4δ < ε. This ends the proof. Theorem 3.3. Let κ≤0,Xa CBB(κ)space, Ya complete CAT(κ)space and A⊆Xnonempty. Then the mapping Φ : N(A, Y )→P(N(X, Y )) is lower semi-continuous. Theorem 3.4. Let Xbe a CBB(0) space, Ya complete CAT(0) space and A⊆Xnonempty. Then the mapping Ψ : L(A, Y )→P(L(X, Y )) is lower semi-continuous. Using the lower semi-continuity of the mappings Φ and Ψ we prove that they admit continuous selections. In order to obtain these singlevalued continuous extension operators we apply a selection result due to Horvath [8] which is a generalization of the classical Michael selection theorem to the setting of c-spaces. Before stating this selection result we recall the following notions: for Za topological space, denote by hZi the family of its nonempty and finite subsets. A mapping F:hZi → P(Z) is a c-structure if firstly, for each A∈ hZi,F(A) is nonempty and contractible, and secondly, for every A1, A2∈ hZi,A1⊆A2implies F(A1)⊆F(A2). The pair (Z, F) is called a c-space and V⊆Zis an F-set if for every A∈ hViwe have that F(A)⊆V. A c-space (Z, F ) is called an l.c. metric space is (Z, d) is a metric space such that open balls are F-sets and if V⊆Zis an F-set, then for every ε > 0, {z∈Z: dist(z, V )< ε}is an F-set. The selection result that we apply is the following. Theorem 3.5 (Horvath [8]).Let Ube a paracompact topological space, (Z, F )an l.c. complete metric space and Γ : U→P(Z)lower semi-continuous such that for each u∈U,Γ(u)is a nonempty and closed F-set. Then there exists a continuous selection for Γ. Let κ≤0. Suppose Xis a CBB(κ) space and Ya complete CAT(κ) space. We check in the sequel that we can indeed make use of the above theorem relying basically on Busemann convexity in Y. We say that B∈P(C(X, Y )) is convex if for every g1, g2∈Band every t∈[0,1] we have that the mapping h:X→Y, h= (1 −t)g1+tg2(that is, h(x) = (1 −t)g1(x) + tg2(x) for every x∈X) belongs to B. Note that balls in C(X, Y ) are convex. The mapping Φ has nonempty and closed values in C(X, Y ). Moreover, for each f∈N(A, Y ), Φ(f) is convex. To see this let f′, f′′ ∈Φ(f) and t∈[0,1]. Then, for each x∈X, dY((1 −t)f′(x) + tf′′(x),(1 −t)f′(y) + tf′′(y)) ≤(1 −t)dY(f′(x), f′(y)) +tdY(f′′(x), f′′(y)) ≤dX(x, y). Similarly, when κ= 0, Ψ is also nonempty, closed and convex-valued. Define F:hC(X, Y )i → P(C(X, Y )) by F(A) = \{B:A⊆B, B convex},for each A∈ hC(X, Y )i. Let A∈ hC(X, Y )i. Then F(A)6=∅. Fix g1∈Aand define H: [0,1] ×F(A)→F(A) by H(t, f) = (1 −t)f+tg1. Note that for each f∈F(A), H(0, f) = fand H(1, f) = g1. It is easy to see that 6 His continuous and so F(A) is contractible. Clearly, for every A1, A2∈ hC(X, Y )i,A1⊆A2implies F(A1)⊆F(A2). Thus, (C(X, Y ), F) is a c-space. Note that a subset of C(X, Y ) is an F-set if and only if it is convex. By Busemann convexity in Yone can finally show that (C(X, Y ), F) is an l.c. metric space. Theorem 3.6. Let κ≤0,Xa CBB(κ)space, Ya complete CAT(κ)space and A⊆Xnonempty. Then there exists a continuous mapping α:N(A, Y )→N(X, Y )such that for all g∈N(A, Y ),α(g)(a) = g(a) for every a∈A. Proof. We can view the mapping Φ with values in P(C(X, Y )) while still preserving its lower semi-continuity. Since any metric space is a paracompact topological space we can now apply Theorem 3.5 to obtain a continuous extension mapping α:N(A, Y )→C(X, Y ). Because Φ actually takes values in P(N(X, Y )) we obtain the conclusion. For bounded Lipschitz mappings we obtain the following result. Theorem 3.7. Let Xbe a CBB(0) space, Ya complete CAT(0) space and A⊆Xnonempty. Then there exists a continuous mapping β:L(A, Y )→L(X, Y )such that for all g∈L(A, Y ),β(g)(a) = g(a)for every a∈Aand Lip(β(g), X) = Lip(g, A). Remark 3.8. Note that in Lemmas 3.1 and 3.2 the lower curvature bound of Xis only used to apply Theorem 2.1. However, when Yis a complete R-tree, one can extend Lipschitz mappings (while keeping the same Lipschitz constant) if Xis an arbitrary metric space. Thus, as before, one can consider even a simpler reasoning in X×Rto obtain that both mappings Φand Ψare lower semi-continuous and admit continuous selections. This property will be improved for the mapping Φin Section 5. Remark 3.9. If κ > 0the argument given in this section does not work in a straightforward way. Note that in this case the direct product X×M2 κis not necessarily a CBB(κ)space. Moreover, the images of the mappings Φand Ψare no longer F-sets when considering the c-structure Fdefined before. 4 A convexity assumption on the images of the extensions In this section we show that one can actually choose extensions in a continuous way even when imposing the condition that the image of the extension belongs to the closure of the convex hull of the image of the original mapping. Related results in the case of Hilbert spaces were recently established in [15], and we extend them to our setting. Recall first the next inequality which stems from the work of Reshetnyak (see, for instance, [9, Theorem 2.3.1] or [18, Lemma 2.1] for a simple proof). Lemma 4.1. Let Ybe a CAT(0) space. Then for every x, y, u, v ∈Y, d(x, y)2+d(u, v)2≤d(x, v)2+d(y, u)2+ 2d(x, u)d(y, v). The property below provides a uniform bound on the distance between the projection points from a common point onto two sets. A similar result in uniformly smooth Banach spaces is [1, Lemma 3.4]. Lemma 4.2. Let Ybe a complete CAT(0) space, C1, C2⊆Ynonempty, closed and convex and suppose r1 and r2are positive numbers. If there exists z∈Ysuch that C1, C2⊆B(z, r1), then for any x∈B(z, r2), d(PC1(x), PC2(x))2≤2(r1+r2)H(C1, C2). Proof. Let C1, C2⊆B(z, r1) and x∈B(z, r2). Denote p1=PC1(x), p2=PC2(x), q1=PC1(p2) and q2=PC2(p1). Clearly, d(p2, q1)≤H(C1, C2) and d(p1, q2)≤H(C1, C2). Note also that that d(x, p1)≤ d(x, z) + d(z, p1)≤r1+r2and d(x, p2)≤r1+r2. Since q1∈C1and p1=PC1(x) it follows that ∠p1(x, q1)≥ π/2 (see [4, Proposition 2.4, page 176]) which yields d(x, q1)2≥d(x, p1)2+d(p1, q1)2. Similarily, d(x, q2)2≥ d(x, p2)2+d(p2, q2)2. By Lemma 4.1 we also have that d(x, q1)2+d(p1, p2)2≤d(x, p2)2+d(p1, q1)2+ 2d(x, p1)d(p2, q1), 7 and therefore d(x, p1)2+d(p1, p2)2≤d(x, p2)2+ 2d(x, p1)d(p2, q1).(2) Likewise, d(x, p2)2+d(p1, p2)2≤d(x, p1)2+ 2d(x, p2)d(p1, q2).(3) Adding (2) and (3) we get that d(p1, p2)2≤d(x, p1)d(p2, q1) + d(x, p2)d(p1, q2)≤2 (r1+r2)H(C1, C2). We prove next a property of the Hausdorff distance. For the corresponding result in the setting of normed spaces, see [19]. Lemma 4.3. Let Ybe a CAT(0) space, C1, C2⊆Ynonempty. Then, H(co(C1),co(C2)) ≤H(C1, C2). Proof. Let c∈C2. Obviously, dist (c, co(C1)) ≤dist (c, C1)≤H(C1, C2). Consider the set E={y∈Y: dist (y, co(C1)) ≤H(C1, C2)}, which is a closed and convex set (by Busemann convexity). Since C2⊆Eit follows that co(C2)⊆E, from where supc∈co(C2)dist (c, co(C1)) ≤H(C1, C2). In a similar way we have that supc∈co(C1)dist (c, co(C2)) ≤ H(C1, C2) and we are done. Theorem 4.4. Let κ≤0,Xa CBB(κ)space, Ya complete CAT(κ)space and A⊆Xnonempty. Then there exists a continuous mapping αc:N(A, Y )→N(X, Y )such that for all g∈N(A, Y ),αc(g)(a) = g(a) for every a∈Aand αc(g)(X)⊆co (g(A)). Proof. By Theorem 3.6 there exists a continuous α:N(A, Y )→N(X, Y ) such that for all g∈N(A, Y ), α(g) extends g. Define a mapping αcon N(A, Y ) by αc(g)(x) = Pco(g(A)) (α(g)(x)) ,for each g∈N(A, Y ) and x∈X. For each g∈N(A, Y ), αc(g)∈N(X, Y ) since the projection onto complete and convex subsets is nonexpansive. Clearly, αc(g)(X)⊆co (g(A)) and αc(g) coincides with gon A. Thus, we only need to prove that αcis continuous. Let f∈N(A, Y ) and ε > 0. Since αis continuous, there exists δ1<1 such that for every g∈N(A, Y ) with d∞(f, g)< δ1we have that d∞(α(f), α(g)) < ε/2. Fix z∈Y. Let r= supx∈XdY(z, α(f)(x)) and take δ= min nδ1,ε2 16(r+1) o.Let g∈N(A, Y ) with d∞(f, g)< δ. Then, for every x∈X, dY(αc(f)(x), αc(g)(x)) = dYPco(f(A)) (α(f)(x)) , Pco(g(A)) (α(g)(x)) ≤dYPco(f(A)) (α(f)(x)) , Pco(g(A)) (α(f)(x)) +dYPco(g(A)) (α(f)(x)) , Pco(g(A)) (α(g)(x)). Note that supa∈AdY(z, f(a)) ≤rand supa∈AdY(z, g(a)) ≤r+ 1. Apply Lemma 4.2 with C1=co (f(A)), C2=co (g(A)) and r1=r2=r+ 1 to get that dYPco(f(A)) (α(f)(x)) , Pco(g(A)) (α(f)(x))≤2√r+ 1pH(co (f(A)) ,co (g(A))) ≤2√r+ 1pH(f(A), g(A)) by Lemma 4.3 ≤2√r+ 1rsup a∈A dY(f(a), g(a)) ≤ε/2. 8 At the same time, dYPco(g(A)) (α(f)(x)) , Pco(g(A)) (α(g)(x))≤dY(α(f)(x), α(g)(x)) ≤d∞(α(f), α(g)) < ε/2. Hence, d∞(αc(f), αc(g)) < ε which proves that αcis continuous too. Following the same idea of proof one can give an analogous result for bounded Lipschitz mappings. Theorem 4.5. Let Xbe a CBB(0) space, Ya complete CAT(0) space and A⊆Xnonempty. Then there exists a continuous mapping βc:L(A, Y )→L(X, Y )such that for all g∈L(A, Y ),βc(g)(a) = g(a)for every a∈A,Lip(βc(g), X) = Lip(g, A)and βc(g)(X)⊆co (g(A)). In fact one can also consider the multivalued extension mappings: •Φc:N(A, Y )→P(N(X, Y )) which assigns to each nonexpansive mapping f∈N(A, Y ) all its nonexpansive extensions f′∈N(X, Y ) with f′(X)⊆co(f(A)). •Ψc:L(A, Y )→P(L(X, Y )) which assigns to each Lipschitz mapping f∈L(A, Y ) all its Lipschitz extensions f′∈L(X, Y ) with Lip(f, A) = Lip(f′, X) and f′(X)⊆co(f(A)). These mappings, too, will be lower semi-continuous. Theorem 4.6. Let κ≤0,Xa CBB(κ)space, Ya complete CAT(κ)space and A⊆Xnonempty. Then the mapping Φc:N(A, Y )→P(N(X, Y )) is lower semi-continuous. Proof. We show that for every f∈N(X, Y ) with f(X)⊆co(f(A)) and for every ε > 0 there exists δ > 0 such that every g∈N(A, Y ) with supa∈AdY(f(a), g(a)) < δ admits an extension g′∈N(X, Y ) with g′(X)⊆co(g(A)) and d∞(f, g′)≤ε. Let fbe as above and ε > 0. By Lemma 3.1 there exists δ > 0 such that every g∈N(A, Y ) with supa∈AdY(f(a), g(a)) < δ admits an extension g1∈N(X, Y ) with d∞(f, g1)≤ε/3. Define g′:X→Y,g′(x) = Pco(g(A)) (g1(x)). Clearly, g′is nonexpansive, extends gand g′(X)⊆co(g(A)). Let x∈X. Then, dY(f(x), g′(x)) ≤dY(f(x), g1(x)) + dY(g1(x), g′(x)) ≤ε/3 + dY(g1(x), g′(x)).(4) For every y∈co(g(A)) we have that dY(g1(x), g′(x)) ≤dY(g1(x), y)≤dY(g1(x), f(x)) + dY(f(x), y), from where dY(g1(x), g′(x)) ≤ε/3 + dist (f(x),co(g(A))) . Consider E={y∈Y: dist (y, co(g(A))) ≤ε/3}. We know that f(A)⊆Esince for any a∈A, dist (f(a),co(g(A))) ≤dY(f(a), g(a)) ≤ε/3. Since Eis closed and convex we have that co(f(A)) ⊆E. But f(X)⊆co(f(A)) and so f(x)∈E. Thus, dist (f(x),co(g(A))) ≤ε/3 which implies that dY(g1(x), g′(x)) ≤2ε/3. Using (4), we obtain that dY(f(x), g′(x)) ≤ε. The same argument yields the result for bounded Lipschitz mappings. Theorem 4.7. Let Xbe a CBB(0) space, Ya complete CAT(0) space and A⊆Xnonempty. Then the mapping Ψc:L(A, Y )→P(L(X, Y )) is lower semi-continuous. Note that one could apply, as in Section 3, Theorem 3.5 to the mappings Φcand Ψcto obtain directly Theorems 4.4 and 4.5, respectively. Remark 4.8. Similar results to the ones given in this section can be proved when Yis a complete R-tree and Xis a general metric space. 9