PROCEEDINGS OF THE AMERICAN MATHEMATICAL Volume K, Number 4. April 1985 SOCIETY CONVEX SETS WITH THE LIPSCHITZ FIXED POINT PROPERTY ARE COMPACT Abstract. Let K be a noncompact convex subset of a normed space X. It is shown that if K is not totally-bounded then there exists a Lipschitz self map / of K with inf{||.v - /(x)||: a- e K) > 0, while if K is totally-bounded then such a map does not exist, but still K lacks the fixed point property for Lipschitz mappings. It follows that a closed convex set in a normed space has the fixed point property for Lipschitz maps if and only if it is compact. 1. Introduction. In [4] Klee proves that a noncompact convex set in a normed space lacks the fixed point property for continuous maps. In this note we extend this result to Lipschitz mappings. Let (K, d) and (S, p) be metric spaces. A function/: K —>S is a Lipschitz map if \\f\\L = sup(- P(f(x),f(z)) -:x,z g K\ < oo. (K,d) has the Lipschitz fixed point property (Li.p.p.) if every Lipschitz self map of K has a fixed point. (K, d) is said to have the approximate Lipschitz fixed point property (approx. L.f.p.p.) if for every Lipschitz self map f of K inf{d(x, f(x)): x g K} = 0. Our main result is the following Theorem 1. Let K be a noncompact convex set in a normed space. (i) If K is not totally-bounded then it lacks the approx. L.f. p. p. (ii) If K is totally-bounded then it has the approx. L.f.p.p., but lacks the L.f.p.p. In [2] Benyamini and the second named author show that the closed unit ball in an infinite dimensional normed space lacks the approx. L.f.p.p. Applying Theorem l(i) to Banach spaces we obtain a more general result. Theorem 2. A closed noncompact convex set in a Banach space lacks the approx. L.f.p.p. Combining Theorem 1 with the Schauder fixed point theorem we obtain Theorem compact. 3. A convex set in a normed space has the L.f.p.p. if and only if it is Received by the editors June 2, 1984. 1980 Mathematics Subject Classification. Primary 47H10. 1Partially supported by NSF grant DMS-8201635. ©1985 American Mathematical Society 0002-9939/85 $1.00 + $.25 per page 633 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 634 P. K. LIN AND Y. STERNFELD In §2 we introduce a metric space A which plays a central role in the proof of Theorem 1, and state 3 propositions which expose the main properties of A. Then we prove Theorem 1. The propositions will be proved in subsequent sections. Through this note the word "space" will always refer to a metric space, and "function" or " map" to Lipschitz mappings, even if not stated explicitly. 2. The space A and proof of Theorem 1. For a set X let lx(X) denote the Banach space of bounded real valued functions on X with the norm \\f\\x = sup{|/(x)|: x G X). In particular for the set N of positive integers lx — lx(N) is the Banach space of bounded real sequences* = (xx, x2, x3,...) with the norm H^H^ = sup„|x„|. For n g N let en = (0,0,..., 0,1,0,0,... A„ = conv{0, en,en + x}, ) G /^ be the « th unit vector, and set n g N; and A= \J A„. »e/V In subsequent sections we shall show that the metric space A (with the metric induced from lx) enjoys the following properties: Proposition 1. The space A as well as the spaces R + = {t g R: / > 0} and (0,l] = {/eA:0</< 1} (with the metric induced from R) are Lipschitz absolute retracts. Definition. A space A' is a Lipschitz absolute retract (L.A.R.) if whenever a space Y contains A as a closed set, there exists a Lipschitz retraction r: Y -» X. A mapping h: K -* S is a Lipschitz equivalence if h is Lipschitz, one-to-one, and hl is Lipschitz. If there exists a Lipschitz equivalence of K onto S then K and 5 are said to be Lipschitz equivalent. Two metric functions d and p on a set A. are equivalent if the identity map id: (K, d) -» (K, p) is a Lipschitz equivalence. Proposition 2. Let K be a noncompact convex subset of a normed space. (i) If K is not totally-bounded then it contains a closed set which is Lipschitz equivalent to either A or R+. More precisely: If some bounded subset of K is not totally-bounded then K contains a closed set which is Lipschitz equivalent to A; while if some ball {x g K: \\x — x0|| < 1} in K is totally-bounded (and K itself is not) then K contains a closed set which is Lipschitz equivalent to R+. (ii) If K is totally-bounded then it contains a closed set which is Lipschitz equivalent to (0,1]. Proposition 3. A lacks the approx. L.f.p.p. Proof of Theorem 1. Theorem l(i), and the second part of Theorem l(ii) follow from Propositions 1, 2, and 3, and the following Lemma. Let (A, d) be a metric space, and let B be a Lipschitz retract of A. If B lacks the L.f.p.p. Proof. (the approx. L.f.p.p.) then so does A. We prove for the approx. L.f.p.p. Let r: A -»fibea retraction, and let /: B -* B be a map with inf{d(x, f(x)): x g B) = a > 0. Let g: A -» A be defined by License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use the lipschitz g = / o r, and e = a/(\\g\\L fixed point property 635 + 2). (Note that ||g||L < ||/||L • \\r\\L.) Let a g A \ B. If d(a, B)> e then d(a, g(a)) > d(a, B) > e. If d(a, B) < e let b g B be such that d(a, b) < e, and then d(a,g(a)) > d(b,g(b))-d(a,b) - d(g(a), g(b)) > a - e - \\g\\Le= e. Hence d(a, g(a)) > e for all aei. For the first part of Theorem l(ii), let ATbe a noncompact totally-bounded convex subset of a normed space X, and let/: K —>K be a Lipschitz map. Let A"denote the completion of X, and A the closure of K in X. Then A' is compact, and / admits an extension /: K —>K. By the Schauder theorem / has a fixed point x0 G K. Let {xn)n^N c A be a sequence which converges to x0. Then {xn) is an approximate fixed point for /, i.e. lim„ d(xn, f(xn)) = 0, and it follows that K has the approx. L.f.p.p. 3. L.A.R.'s, L.A.E.'s, and proof of Proposition 1. (1) Definition. A metric space A is a Lipschitz absolute extensor (L.A.E.) if for every space W, a closed subset Z of W, and a map /: Z -> A, / admits an extension /: W -> X. If there exists a\> 1 such that \\f\\L < a||/||l, then X is said to be a A L.A.E. (2) Example [6, Theorem 1], R is a 1 L.A.E. Proof. Let/: Z -» R be a map, then/(w) an extension of/with ||/||z = ||/||L. (3) Corollary. Proof. = sup{/(z) - ||/||z d(z, w): z g Z} is For every set A, lx(A) is a X L.A.E. Apply (2) to each coordinate/(a,- ), a g A, of a map/: Z -» lx(A). (4) Theorem. A metric space X is a L.A.R. if and only if it is a L.A.E. The following Lipschitz version of a theorem by Hausdorff [3] will be applied in the proof of Theorem 4. Our proof follows that of Arens [1]. (See also [5] for a local version.) (5) Theorem. Let X and W be spaces, Z c W closed, and f: Z -* X a map. There exists a space Y which contains X (isometrically) as a closed set, and an extension g: W -» Yoff. Proof. Note first that Xis isometric to a subset of lx(X). (x '-* d(x,■ ) — d(-, xQ) is an isometry, where x0 G X is some fixed point.) Set B = lx(X) X R, we realize lx(X) in B as lx(X) X {0}, and we may assume that X c lx(X) X {0} c B. So, in particular/: Z -» lx(X) X {0}, and since this is a L.A.E. (by Corollary (3)) / admits an extension h: W -» /^ A) X {0}. Let g: H7 -» fi be defined by g(w) = /¡(w) + (0, d(w, Z)) and set 7 = g(W) U A. One checks easily that X is closed in Y and that g|Z =/. Proof of Theorem 4. L.A.E. => L.A.R. Let A be a L.A.E., and let Y contain A as a closed set. Then an extension r: Y -* X of the identity mapping id: A -> X is a retraction. L.A.R. =» L.^.£. Let Abe a L.A.R., let Z c H/be closed, and let/: Z -> A"be a map. By (5), there exists a space y which contains A as a closed set, and an License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 636 P. K. LIN AND Y. STERNFELD extension g: W -> Y of / Since X is a L.A.R. there exists a retraction Then/= r»g: H7 —>Ais an extension off. (6) Corollary. Proof. r: T —>A. ^ retract of a L.A.R. is a L.A.R. Let A be a retract of a L.A.R. Y with a retraction r: Y -* X. We prove the X is a L.A.E. Let Z c W be closed, and /: Z -* A' be given. Then also f:Z—>Y, and since y is a L.A.E. there exists an extension g: H7 —>Y of / f = r ° g: W -» A is an extension of/. (7) Corollary. Proof. It follows that If Xis Lipschitz equivalent to a L.A.R. then it is a L.A.R. This is trivial for a L.A.E., and hence follows from (4). (8) Proposition. There exists a Lipschitz retraction r: lx -* A. Proof. We consider lx as a lattice with the natural order. Note that for x g A and v*g ¡K 0 < y < x implies v-G A. Let e «=(1,1,1,...) e lx, and x g lx. Set E(x) = {e: e > 0, (x - ee) A 0 G A}. Clearly ||jc|| g E(x). Let e: lx -* R+ be defined by e(x) = inf E(x). Then £ is a Lipschitz map with ||e||z = 1. Indeed, for x and y in lx,x < y + \\x — y\\. Hence e(y) + \\x - y\\ g E(x) and it follows that e(x) < e(y) + \\x - y\\, and by symmetry \e(x) — e(y)\ < \\x - y\\. Let now r: lx -» A be defined by r(x) = (x - e(x) ■e) A 0. Then r is a retraction and \\r\\L < 2. Proof of Proposition 1. The fact that A is a L.A.R. follows from (6), (3), and (8). Since R+ is a retract of R, it is a L.A.R., too. To prove that (0, l] is a L.A.R., we show that [0,1) is a L.A.E. So let Z c W closed and /: Z -» [0,1) be given. Then also/: Z —>[0,1] and since [0,1] is clearly a L.A.E., there exists an extension g: W ~» [0,1] of/. Then 7» 1 = g(w)-—-77-7^: 1 + a( w, Z) W-[0,1) is an extension off. 4. Proof of Proposition space X. We distinguish 2. Let A be a noncompact convex subset of a normed between the following two cases: Case (i): K is not totally-bounded, and Case (ii): K is totally-bounded. Cases (i). Here also we separate the proof into two cases. Cases (i)a. Some bounded subset of K is not totally-bounded. In this case we may assume without loss of generality that 0 g A, and that Kx = {x g K: \\x\\ < 1} is not totally-bounded. Hence, there exists some r > 0 such that Kx cannot be covered by finitely many balls of radius 2r. It follows that 1. For every finite-dimensional linear subspace Y of A, there exists some x g Kx with d(Y, x) ^ r. Indeed, if not then Y + Br d Kx (where Br= {x g A: ||x|| < r}) and from the compactness of {y g Y: \\y\\ < 2} it follows that finitely many balls of radius 2r cover Kx. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 637 THE LIPSCHITZ FIXED POINT PROPERTY Now we select inductively a sequence {xn}n>x in Kx as follows: let xx g Kx be any element with ||xj|| > r. Assume that xx, x2,.. .,xn have been selected. Set Y = span{x,}"=1, and apply 1 to find xn+x g Kx with d(Y, xn+x) > r. For n > 1 set A'„ = conv{0, x„, x„ + 1} and A' = (J A'„. n»l Then A' is a closed subset of A and is Lipschitz equivalent to A by the map h: A -> A' which is defined by h(0) = 0, h(en) = xn, « > 1, and /i is linear on each A„. The verification of this fact is left to the reader. Case (i)b. Some ball {x g A: \\x —x0|| < 1} is totally-bounded. Note that in this case A must be unbounded. Again we assume 0 g A. Let X be the completion of A, and let K be the closure of A in X. Set Kn= {x g K: \\x\\ s£ «}. Then Ä\ is compact, and since (« + l)_1An+1 c n~lKn c Aj, and n~lK„ contains a unit vector for each n, (\„>x n lKn must contain some vectory0 with ||_y0||= 1. Then ty0 g A for all t G A*+. For each « > 1 pick some x„ g A with ||ny0 - x„|| < (n + 10)"1. Then U,?Ssl [xn, xn+x] is a closed subset of A which is Lipschitz equivalent to R+. Case (ii). Once again let A denote the closure of A in the completion of A. Pick x0 g A\A and xx g A. For « > 2 select x„ g A such that ,1-*0 M nI i < 2-c + io) + ~X1 n Then \Jn>x[x„, x,) + 1] is a closed subset of A which is Lipschitz equivalent to (0, l]. 5. A fixed point free map on A. In this section we construct a Lipschitz map /: A -» A without an approximate fixed point, i.e. inf{||jc —f(x)\\: x g A} > 0. For x = (xx, x2,...) g A, let \\xx\\ = Z^,|x,| denote the lx norm of x. Note that the metric functions induced on A by ||x||i and H-xjl^ = sup,|x,| are Lipschitz equivalent, and we apply the lx norm in the construction off. First we define/on 3A = {a: g A: \\x\\x < \) as follows: /(0) = ex,f(\ex) = ex,f(\en) = 0 for n > 2, and/is linear on }A„ for each n > 1. Note that f(\A) = [0, ex], and that forx g ^A,,, n > 2, with \\x\\x = }/(jc) = 0. As a piecewise linear map,/is Lipschitzian on }A. Next, we define /on {x g A: ||x||, 5= f}; / will map this set onto {x g A: ||x||, = 1}. Letg: {x g A: ||x||i = 1} -» [1, 00) = {r g R: r ^ 1} be defined by g((l - t)en+te„ + x)) = n + t, n>l,0</<l. g is one-to-one and onto. We shall construct a map h: {x g A: ||x||j ^ f} -» [l,oo) and/will be defined to be/ = g"1 ° /i. h is defined as follows: Forx g A, ||x||, = f3> /i(jc) -^IttVI +' T' ^GAi' llvll, 2' *«1 X \ 1 ^||l/ L x^ IJA„. »>2 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 638 P. K. LIN AND Y. STERNFELD X=e, ek«flM«fGH0*fÜW xe A{£wá dose to e, ffo) fi*i xe^i omc( close to e* fY°) X= e* &=f(x) 0= ffiAj XeArx n>2 e* * Figure 1 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use e«« THE LIPSCHITZ FIXED POINT PROPERTY 639 And for x g A, \\x\\x = 1, h(x) = g(x) + X. For x g A with § < \\x\\x, h is defined to be linear on [f (x/HxHj), x/HxIlJ. Under the metric d(r, s) = \r — s\ A X on [0,1), g is a Lipschitz equivalence, and h is Lipschitz. Hence/ = g"1 » h is Lipschitz. To extend / to the whole of A, note that for 0 # x G A, /(y(*/ll*lli)) and /(§ (x/||x||i)) are in the same A„ for some n > 1. Indeed, if x g Ax then both are in Ax, while if x <£ Ax then/(}(x/||x||1)) = 0. Hence we extend /by letting it be linear on [\x, fx] for all x g A, Hjc]^ = 1. The extended map is still Lipschitzian. Figure 1 illustrates [0, x] and/([0, x]) for several values of x g A with ||x||, = 1. For x g A with \\x\\x = 1, [0, x] n/([0, x]) Ç {0, x}, and since neither 0 nor x (with \\x\\x = 1) are fixed points of/,/is fixed point free. Hence, by compactness, inf{||x -/(x)||: ||tx —/(tx)||, x g Ax U A2} = a > 0. But for x g A,„ n > 2. \\x - f(x)\\ = where t: A„ -» A2 is the natural isometry. It follows that \\x - f(x)\\ > a for all x G A. References 1. R. Arens, Extension of functions on fully normal spaces, Pacific J. Math. 2 (1952), 11-22. 2. Y. Benyamini and Y. Sternfeld, Spheres in infinite dimensional normed spaces are Lipschitz contracl- ible, Proc. Amer. Math. Soc. 88 (1983),439-445. 3. F. Hausdorff, Erweiterung einer stetigen Abbildung, Fund. Math. 30 (1938), 40-47. 4. V. Klee, Some topological properties of convex sets. Trans. Amer. Math. Soc. 78 (1955), 30-45. 5. J. Luukkainen, Extension of spaces, maps and metrics in Lipschitz topology, Ann. Acad. Sei. Fenn. Ser. A 17 (1978). 6. E. J. McShane, Extension of ranges of functions. Bull. Amer. Math. Soc. 40 (1934), 837-842. Department of Mathematics, The University Department of Mathematics, University Department of Mathematics, 90007 of Southern The University Current address (P. K. Lin): Department of Texas at Austin, Austin, Texas 78712 California, Los Angeles, California of Haifa, Haifa, Israel of Mathematics, University of Iowa, Iowa City, Iowa 52242 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use