Smooth Lipschitz retractions of starlike bodies onto their boundaries in infinite-dimensional Banach spaces

SMOOTH LIPSCHITZ RETRACTIONS OF STARLIKE BODIES ONTO THEIR BOUNDARIES IN INFINITE-DIMENSIONAL BANACH SPACES DANIEL AZAGRA AND MANUEL CEPEDELLO BOISO Abstract. Let Xbe an infinite-dimensional Banach space and let Abe a Cp Lipschitz bounded starlike body (for instance the unit ball of a smooth norm). We prove that (1) The boundary ∂A is CpLipschitz contractible. (2) There is a CpLipschitz retraction from Aonto ∂A. (3) There is a CpLipschitz map T:A−→ Awith no approximate fixed points. 1. Introduction and main results The well known Brouwer’s fixed point theorem states that every continuous selfmap of the unit ball of a finite-dimensional Banach space admits a fixed point. This is equivalent to saying that there is no continuous retraction from the unit ball onto the unit sphere, or that the unit sphere is not contractible (the identity map on the sphere is not homotopic to a constant map). This result is no longer true in infinite dimensions (see [8]). In [14] B. Nowak showed that for several infinite-dimensional Banach spaces Brouwer’s theorem fails even for Lipschitz mappings, and in [6] Y. Benyamini and Y. Sternfeld generalized Nowak’s result for all infinite-dimensional normed spaces, establishing that for every infinite-dimensional space (X, k·k) there exists a Lipschitz retraction from the unit ball BX={x∈X:kxk ≤ 1}onto the sphere SX={x∈X:kxk= 1}, and that SXis Lipschitz contractible. In recent years a lot of work has been done on smoothness and Lipschitz properties in Banach spaces (see [11, 5]). Following this trend it is natural to ask whether Nowak-Benyamini-Sternfeld’s results can be sharpened so as to get Cpsmooth Lipschitz retractions of the unit ball onto the sphere of every infinite-dimensional Banach space whose norm is Cpsmooth. In this note we will show that this is indeed possible. In fact we generalize those results in two ways. Not only do they hold for the smooth category but also for a wider class of objects than balls and spheres: we show that for every infinite-dimensional Banach space with a CpLipschitz bounded starlike body A(where p= 0,1,2, . . . , ∞), there is a CpLipschitz retraction of A onto its boundary ∂A, and ∂A is also CpLipschitz contractible. At this point we need to introduce some terminology. A closed subset Aof a Banach space Xis said to be a starlike body provided Ahas a non-empty interior and there exists a point x0∈intAsuch that each ray emanating from x0meets the boundary of Aat most once. In this case we will say that Ais starlike with respect to x0. When dealing with starlike bodies, we can always assume that they are starlike 1991 Mathematics Subject Classification. Primary: 46B20. Secondary: 58B05, 46T05. 1 2 DANIEL AZAGRA AND MANUEL CEPEDELLO BOISO with respect to the origin (up to a suitable translation), and we will do so unless otherwise stated. For a starlike body A, we define the Minkowski functional of Aas qA(x) = inf{λ > 0|1 λx∈A} for all x∈X. It is easily seen that for every starlike body Aits Minkowski functional qAis a continuous function which satisfies qA(rx) = rqA(x) for every r≥0. Moreover, A={x∈X|qA(x)≤1}, and ∂A ={x∈X|qA(x) = 1}, where ∂A stands for the boundary of A. Conversely, if ψ:X−→ [0,∞) is continuous and satisfies ψ(λx) = λψ(x) for all λ≥0, then Aψ={x∈X|ψ(x)≤1}is a starlike body. Convex bodies (that is, closed convex sets with nonempty interior) are an important kind of starlike bodies. We will say that Ais a Cpsmooth (Lipschitz) starlike body provided its Minkowski functional qAis Cpsmooth (and Lipschitz) on the set X\q−1 A(0). Smooth starlike bodies are interesting because they are strongly related to bump functions and to n-homogeneous polynomials in Banach spaces (see [2] and [3]), therefore their geometrical properties are worth studying. It is worth noting that every Banach space having a Cpsmooth (Lipschitz) bump function has a Cpsmooth (Lipschitz) bounded starlike body too (and the converse is also true). Before stating our main result we need a few topological definitions. Let M,N be closed subsets of a Banach space X. We will say that two maps f, g :M−→ N are CpLipschitz homotopic provided there exist an open subset Uof Xcontaining M, an ε > 0, and a Cpsmooth mapping H: (−ε, 1 + ε)×U−→ Xsuch that the restriction of Hto [0,1] ×Mis a Lipschitz homotopy joining fto g, that is, H: [0,1] ×M−→ Nis Lipschitz continuous and satisfies H(0, x) = f(x) and H(1, x) = g(x) for all x∈M. Moreover we will demand that H(t, x) = f(x) for t≤0, x∈M, and H(t, x) = g(x) for t≥1, x∈M. It is not difficult to see that, with this definition, ‘being CpLipschitz homotopic’ endows the set of CpLipschitz mappings from Minto Nwith an equivalence relationship (one can join Cpsmooth homotopies without losing smoothness or Lipschitzness). A closed subset Mof Xis said to be CpLipschitz contractible if the identity map on Mis CpLipschitz homotopic to a constant map on M. For instance, it is easy to check that every CpLipschitz starlike body Ais CpLipschitz contractible. It is also easy to see that every two maps on a (CpLipschitz) contractible set are always (CpLipschitz) homotopic (they are both homotopic to a constant). Finally, we will say that r:A−→ ∂A is a Cpsmooth Lipschitz retraction from the starlike body Aonto its boundary provided there exist an open subset Uof X containing Aand a Cpsmooth mapping R:U−→ Xsuch that Rfixes all the points of ∂A, and the restriction of Rto Ais Lipschitz continuous and coincides with r. Our main result is the following Theorem 1.1. Let Xbe an infinite-dimensional Banach space and let Abe a Cp Lipschitz bounded starlike body. Then: (1) The boundary ∂A is CpLipschitz contractible. (2) There is a CpLipschitz retraction from Aonto ∂A. SMOOTH LIPSCHITZ RETRACTIONS OF STARLIKE BODIES ONTO THEIR BOUNDARIES 3 (3) There is a CpLipschitz map T:A−→ Awith no approximate fixed points, that is, inf{kx−T(x)k:x∈A}>0. As a corollary we obtain the following generalization of Benyamini-Sternfeld’s theorem: Corollary 1.2. Let (X, k·k)be an infinite-dimensional Banach space with an equivalent norm k·kwhich is Cpsmooth, and let BXand SXbe its unit ball and unit sphere respectively. Then (1) SXis CpLipschitz contractible. (2) There is a CpLipschitz retraction of BXonto SX. (3) There is a CpLipschitz map T:BX−→ BXwith no approximate fixed points. If one is not interested in the Lipschitz property, it is a trivial consequence of the main result in [1] (see also [4]) that the sphere SXis Cpcontractible and there are Cp smooth retractions from BXonto SX. Unfortunately, the deleting diffeomorphisms obtained in [1, 4] are not Lipschitz, and corollary 1.2 cannot be deduced using those results. As a matter of fact, corollary 1.2 provides a new result even in the case X=`2with the usual hilbertian norm. 2. The proofs The proof of the main result is rather technical and will be split into three propositions and several lemmas. The general scheme of the proof follows that of [6], which in turn is a generalization with some modifications of Nowak’s approach [14]. The proofs in [6, 14] are already involved in themselves and here they will be complicated with the difficulties peculiar to smooth maps and starlike bodies. First of all it should be noted that parts (2) and (3) of theorem 1.1 are straightforward consequences of (1). Indeed, assume that ∂A is CpLipschitz contractible. Then there are an open subset Uof Xcontaining ∂A and a Cpsmoth map H: (−ε, 1 + ε)×U−→ Xsuch that the restriction of Hto [0,1] ×∂A is a Lipschitz homotopy joining the identity to a constant x0in ∂A. Without loss of generality we may assume that His defined on (−∞,+∞)×Uand has the property that H(t, x) = x0for all t≤0, x∈∂A, and H(t, x) = xfor all t≥1, x∈∂A. Then the formula R(x) = H(2ψ(x)−1,x ψ(x)), where ψis the Minkowski functional of A, defines a Cpsmooth map on X\{0}with the property that R(x) = x/ψ(x) whenever ψ(x)≥1 and R(x) = x0if ψ(x)≤1 2. Then one can obviously extend R(by putting R(0) = x0) to a Cpsmooth map R:X−→ Xsuch that R(x) = xwhenever ψ(x) = 1 and R(x) = x0for ψ(x)≤1 2. The restriction of Rto the set A={x∈X:ψ(x)≤1}gives us a Cpsmooth retraction rfrom Aonto its boundary. By using the fact that H: [0,1]×∂A −→ ∂A is Lipschitz, it is easily seen that r:A−→ ∂A is Lipschitz as well. This shows that part (1) of the theorem implies (2). On the other hand, once we have such a Cp Lipschitz retraction rone can easily get a CpLipschitz map T:A−→ Awith no approximate fixed points: it is enough to take T(x) = −r(x). 4 DANIEL AZAGRA AND MANUEL CEPEDELLO BOISO Let us now start the proof of part (1) of 1.1. The following lemma tells us that for every two CpLipschitz bounded starlike bodies A1, A2the pair (A1, ∂A1) is Cp Lipschitz equivalent to the pair (A2, ∂A2). We omit the proof of this result since it is an easy adaptation of that of Proposition 3 in [2]. Lemma 2.1. Let Xbe a Banach space, and let A1, A2be CpLipschitz bounded starlike bodies. Then there exist a Cpbi-Lipschitz diffeomorphism g:X−→ Xsuch that g(A1) = A2,g(∂A1) = ∂A2, and g(0) = 0. Moreover, g(x) = µ(x)x, where µ:X−→ [0,∞), and hence gpreserves the rays emanating from the origin. Therefore, any CpLipschitz property of a bounded starlike body or its boundary is shared with all the bounded starlike bodies and their boundaries. In particular the main theorem and all the auxiliary results which we will introduce in this section can be proved for any particular CpLipschitz bounded starlike body in a Banach space Xand then, by using this lemma, extended for the rest of CpLipschitz bounded starlike bodies, which are all equivalent. We will use this fact later on without further notice. We will also need the following technical definition. Definition 2.2. Let Xbe a Banach space with a Cpsmooth Lipschitz bounded starlike body A, and let ψbe its Minkowski functional. Let Mbe a closed subset of X,y0∈M, and ε > 0. For every y∈M,δ > 0, define the pseudoball Bψ M(y, δ) = {x∈M:ψ(x−y)≤δ}. A point y0is said to be an ε-escaping point for ψin Mprovided there exists a Cpsmooth Lipschitz mapping T:M−→ Msatisfying: (1) Tis Lipschitz homotopic to the identity on M. (2) inf{ψ(Tny0−Tmy0) : n>m≥0} ≥ 10ε. (3) For all n≥0,Tmaps Bψ M(Tny0,2ε)isometrically onto Bψ M(Tn+1y0,2ε) and, moreover, Tis merely a traslation when restricted to these sets. (4) For all n≥0,T−1(Bψ M(Tn+1y0,2ε)) = Bψ M(Tny0,2ε). Now we state the three auxiliary propositions that we will use in the proof of the main theorem. Proposition 2.3. Let M,Nbe closed subsets of a Banach space Xwhich has a CpLipschitz bounded starlike body Awith Minkowski functional ψ. Suppose there is an ε-escaping point y0in M. Let g: [−1,1] ×M−→ Nbe a CpLipschitz map which constantly attains the value z0∈Noutside the set [1 4,3 4]×Bψ M(y0, ε). Assume moreover that there exists an open subset Uof Xcontaining Mand an extension g: (−1−ε, 1 + ε)×U−→ Xof gsuch that gis Cpsmooth and satisfies g(t, x) = z0 for all t∈(−1−ε, 1+ε)and x /∈Bψ U(y0, ε). Then gis CpLipschitz homotopic to the constant function z0in [−1,1] ×Mby means of a CpLipschitz homotopy Hτ(t, x) (0≤τ≤1,(t, x)∈[−1,1] ×M) for which Hτ(t, x) = z0whenever |t| ≥ 3 4. Proposition 2.4. Let Xbe an infinite-dimensional Banach space with a CpLipschitz bounded starlike body. Then there exist ε > 0and another CpLipschitz bounded symmetric starlike body Wsuch that its boundary ∂W has an ε-escaping point with respect to ψ=qW, the Minkowski functional of W. SMOOTH LIPSCHITZ RETRACTIONS OF STARLIKE BODIES ONTO THEIR BOUNDARIES 5 Proposition 2.5. Let Xbe a Banach space and let Abe a CpLipschitz starlike body which is bounded and symmetric, x0∈∂A,ε > 0. Then the identity map on ∂A is CpLipschitz homotopic to a map f:∂A −→ ∂A which constantly attains the value −x0outside the set {x∈∂A :ψ(x−x0)< ε}(where ψis the Minkowski functional of A). Moreover, fcan be assumed to have a Cpsmooth extension f:U−→ X (where Uis an open subset of Xcontaining ∂A) such that f(x) = −x0whenever ψ(x−x0)≥ε,x∈U. Proof of the theorem. Let Ybe a closed hyperplane of X. By Proposition 2.4 there is a CpLipschitz bounded symmetric starlike body Won Ysuch that its boundary ∂W admits an ε-escaping point y0, for some ε > 0 that can be assumed to satisfy 0 < ε < 1 4. Let qWbe the Minkowski functional of this starlike body. We may write X=R×Y. Now, let Vbe a C∞smooth Lipschitz bounded symmetric convex body of the plane R2such that its boundary ∂V contains the set {(t, s)∈R2:|t| ≤ 1,|s|= 1}, and consider the Minkowski functional qVof V, which is a C∞smooth equivalent norm on R2. Define now ψ(t, y) = qV(t, qW(y)) for every (t, y)∈R×Y=X. It is clear that ψis a CpLipschitz function on X\{0} which is symmetric and positive homogeneous. Then U={(t, y)∈X:ψ(t, y)≤1} is a CpLipschitz bounded symmetric starlike body with the property that its boundary ∂U contains the band [−1,1] ×∂W . Without loss of generality we can assume that U=A(see Lemma 2.1 and the preceding remarks), and it suffices to prove the theorem for this particular starlike body. Next put x0= (1 2, y0)∈∂A and z0=−x0. By Proposition 2.5 there exists a CpLipschitz map f:∂A −→ ∂A which is CpLipschitz homotopic to the identity on ∂A, and which has a Cpsmooth extension f:U−→ Xsuch that f(x) = −x0whenever ψ(x−x0)≥ε,x∈U. Note that if x= (t, y)∈∂A satisfies ψ(x−x0)< ε then, since ε < 1 4, and taking into account the particular shape of ∂A, we have that (t, y)∈[1 4,3 4]×Bψ ∂W (y0, ε)⊂[−1,1] ×∂W. Then it is clear that g=f|[−1,1]×∂W satisfies the conditions of Proposition 2.3 with M=∂W (bear in mind that g(t, y) = f(t, y) = z0whenever ψ(0, y −y0)≥εbecause the pseudoball {x∈X:ψ(x−x0)< ε}is contained in the cilynder {(t, y)∈X:qW(y−y0)< ε}). Since y0is an ε-escaping point in ∂W, it follows that gis CpLipschitz homotopic, as a map from [−1,1] ×∂W into ∂A, to the constant z0=−x0∈∂A, by a Cp Lipschitz homotopy Hτ(t, y) satisfying Hτ(t, y) = z0whenever |t| ≥ 3 4. Now, from the particular construction of ∂A, it is clear that one can extend Hτ to a CpLipschitz homotopy Fτby defining Fτ(x) = z0for x∈∂A \([−1,1] ×∂W ), and it is easily checked that Fτis a CpLipschitz homotopy joining fto the constant z0in ∂A. Since fis itself CpLipschitz homotopic to the identity on ∂A, we can conclude that ∂A is CpLipschitz contractible to a point. 6 DANIEL AZAGRA AND MANUEL CEPEDELLO BOISO Now we will give the proofs of Propositions 2.3, 2.4 and 2.5. Proof of Proposition 2.3. Let Tbe the map associated to ψand the ε-escaping point y0in Definition 2.2. Let θ:R−→ [0,∞) be a CpLipschitz mapping such that θis strictly increasing in (0,∞), θ(−t) = θ(t), θ(0) = 0, and θ(t) = |t|for |t| ≥ 1 8. Pick another nondecreasing Cpmap ζ:R−→ Rsuch that ζ(t) = 0 for t≤1 4and ζ(t) = 1 for t≥3 4. Now let us define two maps f0, f1: [−1,1] ×M−→ Nby f0(t, x) =    g(θ(t), T−n(x)) whenever t≥0, x ∈Bψ M(Tn(y0), ε), and n≥0; g(θ(t), T−n(x)) whenever t≤0, x ∈Bψ M(Tn(y0), ε), and n≥1; z0otherwise; and f1(t, x) = g(θ(t), T−n(x)) whenever t≥0, x ∈Bψ M(Tn(y0), ε), and n≥0; z0otherwise. Note that on each “rectangle” [1 4,3 4]×Bψ M(Tn(y0), ε) or [−3 4,−1 4]×Bψ M(Tn(y0), ε), n≥0, the mappings f0and f1are defined by the corresponding value (with respect to T−n) of gin the rectangle [1 4,3 4]×Bψ M(y0, ε). All these rectangles are disjoint, by the definition of ε-escaping point. Since T−nis merely an affine traslation of Bψ M(Tn(y0),2ε) onto Bψ M(y0,2ε) and gis Cpsmooth and Lipschitz, it is clear that the maps f0, f1are Cpand Lipschitz as well. By assumption, Tis CpLipschitz homotopic to the identity; let Gτ, τ ∈[0,1], be a CpLipschitz homotopy joining the identity to Tin M. Then Fτ(t, x) = f0(t, x) for t≥0; f0(t, Gτ(x)) for t≤0, is a CpLipschitz homotopy joining f0to f1in [−1,1]×M. Now, the map F1 τ(t, x) = f1(θ(t)(1−ζ(τ))+ζ(τ), x) defines a CpLipschitz homotopy joining f1to the constant z0, and it is not difficult to see that the map F0 τ(t, x) = (f0(θ(t)ζ(τ) + (1 −ζ(τ)), x) for x /∈Bψ M(y0, ε); g(t, x) for x∈Bψ M(y0, ε) is a CpLipschitz homotopy joining gto f0(here we use the fact that g(t, x) = z0 whenever ψ(x−x0)≥ε,x∈U). We can then obtain the desired homotopy Hτ by applying successively F0 τ,Fτand F1 τ. Since all of these homotopies have the constant value z0for |t| ≥ 3 4, the same is true of Hτ. In order to prove Proposition 2.4 a number of rather technical lemmas and facts will be required. Let us fix some standard notation used throughout these statements. If Kis a subset of Xand x∈X, we denote by dψ(x, K) := inf{ψ(x−y) : y∈K}. Also, the closed starlike body Bψ X(x, r) = {y∈X:ψ(y−x)≤r}will be simply written as Bψ(x, r). The first technical tool we need is somehow a smooth version of Uryshon’s lemma. SMOOTH LIPSCHITZ RETRACTIONS OF STARLIKE BODIES ONTO THEIR BOUNDARIES 7 Lemma 2.6. Let Xbe a Banach space, and let Abe a CpLipschitz bounded symmetric starlike body with Minkowski functional ψand Kbe a compact subset of X. Then, for every r > 0there exists a CpLipschitz function f=fψ,r,K :X−→ [0,1] such that (1) f(x) = 1 whenever dψ(x, K)≤r/2, and (2) f(x) = 0 whenever dψ(x, K)≥r. Proof. Let Lψbe the Lipschitz constant of ψ(i.e.,ψ(x)−ψ(y)≤Lψkx−yk, for all x, y ∈X). Since Kis compact there exist x1, . . . , xl∈Ksuch that K⊂ l [ j=1 Bk·kxj,r 4Lψ.(∗) Then pick a non-decreasing C∞function g:R−→ [0,1] such that g−1(0) = (−∞,3 4r] and g−1(1) = [7 8r, ∞). Put h(x) = l Y j=1 g(ψ(x−xj)) for all x∈X. Since the functions x7→ g(ψ(x−xj)) are all bounded, Lipschitz and Cp, the function h, being a finite product of such functions, is also Cpsmooth and Lipschitz. Moreover, note that the Lipschitz constant of honly depends on ψ,rand the number of elements of the covering (∗). By the construction of hit is quite clear that h(x) = 1 if x /∈ ∪l j=1Bψ(xj,7 8r), and therefore h(x) = 1 whenever dψ(x, K)≥r. Moreover, is easy to see that h(x) = 0 if x∈G:= ∪l j=1Bψ(xj,3 4r). Let us check that G⊇ {x∈X:dψ(x, K)≤r/2}. In fact, if x∈Xis such that dψ(x, K) = r/2, take y∈Kin such a way that ψ(x−y) = r/2 and xjso that y∈Bk·k(xj,r 4Lψ). Then it follows ψ(x−xj)≤ψ(x−y) + Lψky−xjk ≤ 3 4r. In order to conclude the proof it suffices to take f(x) = 1 −h(x).  Fact 2.7. Let Xbe a Banach space which has a CpLipschitz bounded symmetric starlike body Awith Minkowski functional ψand r > 0. Then for some M > 0 one has that for every a, b with kak=kbk=1 4there exists a CpLipschitz function fa,b :X−→ [0,1] whose Lipschitz constant is less than or equal to M, and which satisfies that (1) fa,b(x) = 1 whenever dψ(x, [a, b]) ≤r/2, and (2) fa,b(x) = 0 whenever dψ(x, [a, b]) ≥r. Proof. Fix r > 0. For every two arbitrary points a, b of Xsatisfying kak=kbk=1 4, consider the compact set K= [a, b]. From Lemma 2.6, there exists a function fa,b that verifies conditions (1) and (2). We only have to ensure that the function fa,b constructed in the proof of Lemma 2.6 can be chosen with a Lipschitz constant that does not depend on the segment [a, b]. As we remarked before, the Lipschitz constant of fa,b only depends on the number of elements of the finite covering chosen in (∗). But, since the diameter of any segment [a, b] is uniformly bounded, for every 8 DANIEL AZAGRA AND MANUEL CEPEDELLO BOISO pair aand bit is easy to find an appropriate covering of [a, b] with a fixed number of elements.  Lemma 2.8. Let Xbe a Banach space which has a CpLipschitz bounded symmetric starlike body Awith Minkowski functional ψ. Then for every r > 0there exists a constant L > 0so that for every a, b ∈Xwith kak=kbk=1 4there is a map F=Fa,b :A−→ Asatisfying (1) Fis CpLipschitz, and the Lipschitz constant of Fis less than or equal to L (and therefore only depends on ψand r, but not on a, b). (2) Fmaps Bψ(a, r/2) isometrically onto Bψ(b, r/2); in fact Fis merely a translation when restricted to these sets, and F(a) = b. (3) F−1(Bψ(b, r/2)) = Bψ(a, r/2). (4) F(x) = xwhenever dψ(x, [a, b]) ≥r. (5) Fmaps lines parallel to the segment [a, b]into themselves. Proof. For every such a, b let us define F=Fa,b :X−→ Xby F(x) = x+fa,b(x)(b−a) for all x∈X, where fa,b is the corresponding function obtained from fact 2.7. It is clear that Fis Cpsmooth and Lipschitz on X, with a Lipschitz constant not greater than L=M+ 1. Therefore Fsatisfies condition (1) of the lemma. It is evident from the definitions of Fand fa,b that Fsatisfies properties (2), (4) and (5) as well. Let us see that Fsatisfies property (3). If F(x)∈Bψ(b, r/2) then r 2≥ψ(b−F(x)) = ψ(fa,b(x)a+ (1 −fa,b(x))b−x). Since 0 ≤fa,b(x)≤1 we have fa,b(x)a+ (1 −fa,b(x))b∈[a, b] and, therefore, it follows that dψ(x, [a, b]) ≤r/2 and fa,b(x) = 1. Henceforth, we have r 2≥ψ(b−F(x)) = ψ(b−(x+ (b−a)) = ψ(a−x), which means that x∈Bψ(a, r/2). This shows that F−1(Bψ(b, r/2)) = Bψ(a, r/2).  Lemma 2.9. Let Xbe an infinite-dimensional Banach space which has a CpLipschitz bounded symmetric starlike body Awith Minkowski functional ψ. Then there exist some ε > 0and a point x0in the interior of Awhich is an ε-escaping point in Awith respect to a map T:A−→ Awhich in addition to properties (1)–(4) of Definition 2.2 satisfies T(x) = xwhenever ψ(x)≥3 4. Proof. Without loss of generality we can assume that BX⊆A. Since Ais a bounded starlike body we know that there exists some α > 0 such that αkxk ≤ ψ(x)≤ kxkfor all x∈X. Note that no matter how Tis defined, Twill be CpLipschitz homotopic to the identity on Abecause Ais starlike and hence CpLipschitz contractible (so that both Tand the identity are homotopic to a constant on A). Let (wn)n∈Nbe a normalized basic sequence in Xwith biorthogonal functionals (w∗ n)n∈N⊂X∗satisfying kw∗ nk ≤ 4 (one can always take such a sequence, see [10], p. 93), and put zn=1 4wnfor all n∈N. Let us denote by Ln,k the straight line {tzn+ (1 −t)zk:t∈R}passing through znand zk(for n6=k). It is easy to see SMOOTH LIPSCHITZ RETRACTIONS OF STARLIKE BODIES ONTO THEIR BOUNDARIES 9 that, if {n, k}∩{m, l}=∅then kx−yk ≥ 1 32 for all x∈Ln,k, y ∈Lm,l. This implies that ψ(x−y)≥α 32 for all x∈Ln,k, y ∈Lm,l; that is, dψ(Ln,k, Lm,l)≥α 32 . Now take r=α 320 , and for every n, k ∈N,n6=k, pick a function Fn,k :A−→ A satisfying the conditions of Lemma 2.8 for a=znand b=zk, and put ε=r/4. For this choice of εand rwe have dψ(Ln,k, Lm,l)≥α 32 ≥10r > 20ε. (∗∗) Note that, by this inequality and the construction of F, if {n, k}∩{m, l}=∅then Fn,k(x) = xwhenever dψ(x, Lm,l)≤r= 4ε, and in particular whenever Fm,l(x)6=x or x=Fm,l(y) for some y6=x. Then the infinite composition V1(x) = (· · · ◦ F2n−1,2n◦ · · · ◦ F3,4◦F1,2)(x) is well defined and satisfies (1) V1is Lipschitz. Indeed, take into account that all the maps Fn,k involved in the definition of V1have a Lipschitz constant which is less than or equal to a fixed constant L, and the infinite composition defining V1is uniformly locally finite. In fact for every x∈Athere exists a neighbourhood of xin Asuch that V1coincides with one of the Fn,k when restricted to this neighbourhood. From these properties and from the facts that Acontains the unit ball BX, which is a convex set, and V1obviously restricts to the identity outside BX, one can easily deduce that V1is Lipschitz (with a Lipschitz constant less than or equal to L) on A. (2) V1is Cpsmooth (this is again a consequence of the fact that the infinite composition defining V1is locally finite and all the functions Fn,k are Cp smooth). (3) V1maps Bψ(z2n−1,2ε) isometrically (in fact it is merely a translation when restricted to this set) onto Bψ(z2n,2ε). (4) V−1 1(Bψ(z2n,2ε)) = Bψ(z2n−1,2ε). (5) V1(x) = xwhenever ψ(x)≥3 4. Let us define as well V2(x) = (· · · ◦ F2n,2n+1 ◦ · · · ◦ F4,5◦F2,3)(x). Then V2is also a CpLipschitz map that satisfies (3’) V2maps Bψ(z2n,2ε) isometrically (in fact it is a translation when restricted to this set) onto Bψ(z2n+1,2ε). (4’) V−1 2(Bψ(z2n+1,2ε)) = Bψ(z2n,2ε). (5’) V2(x) = xwhenever ψ(x)≥3 4. Now let us define T=V2◦V1. It is clear that Tis a CpLipschitz map. It only remains to check that z1is an ε-escaping point for T. Indeed, as said above, condition (1) of Definition 2.2 is trivially satisfied. It is also clear that Tnz1=z2n+1, and condition (2) of 2.2 follows from (∗∗) above. Finally, conditions (3) and (4) of 2.2 follow respectively from (3, 3’) and (4, 4’) above.  Proof of Proposition 2.4.