Approximation of Lipschitz functions by Δ-convex functions in banach spaces

arXiv:math/9702212v1 [math.FA] 13 Feb 1997 APPROXIMATION OF LIPSCHITZ FUNCTIONS BY ∆-CONVEX FUNCTIONS IN BANACH SPACES Manuel Cepedello Boiso February 13, 1997 Abstract. In this paper we give some results about the approximation of a Lipschitz function on a Banach space by means of ∆-convex functions. In particular, we prove that the density of ∆-convex functions in the set of Lipschitz functions for the topology of uniform convergence on bounded sets characterizes the superreflexivity of the Banach space. We also show that Lipschitz functions on superreflexive Banach spaces are uniform limits on the whole space of ∆-convex functions. 0. Introduction and Notations A function defined on a Banach space Xis called ∆-convex if it can be expressed as a difference of continuous convex functions or, equivalently, if it belongs to the linear span of the continuous convex functions on X. The purpose of this paper is to give some necessary and sufficient conditions for the approximation of Lipschitz functions by ∆-convex functions. The initial motivation for our work comes from two recent articles of R. Deville, V. Fonf and P. H´ajek ([DFH1] and [DFH2]). A consequence of their results is that, under certain conditions on the Banach space X, any convex function on Xwhich is bounded on bounded sets can be approximated by smooth convex functions. It is therefore natural to consider the class of Banach spaces for which the ∆-convex functions are dense in the class of Lipschitz functions in order to extend this property of smooth approximations. Our main result is the following characterization of superreflexivity. Theorem 0. Let Xbe a Banach space. Then Xis superreflexive if and only if every Lipschitz function on Xcan be approximated uniformly on bounded sets by differences of convex functions on Xwhich are bounded on bounded sets. A consequence of Theorem 0 is that the above mentioned approach does not provide any new result, because it works only for superreflexive spaces (for which the smooth approximation property is known using the existence of partitions of unity, see Ch. VIII of [DGZ]; for the analytic approximation case, see [K]). For the superreflexive case, we give explicit formulas for the ∆-convex approximation of a Lipschitz function. These formulas, which are simpler than those from 1991 Mathematics Subject Classification. Primary 46B20; Secondary 46B10. Key words and phrases. Convex functions, superreflexivity in Banach spaces. The author was supported by a FPU Grant of the Spanish Ministerio de Educaci´on y Ciencia. Typeset by A M S-T EX [St], also provide uniform convergence on the whole space X. Specifically, the degree of convergence given by our formulas relies directly on the rotundity of the equivalent norm that it is used. Similar ideas in this direction can be found in [A] and [PVZ]. We thank P. H´ajek for bringing to our attention the link between our work and the distortion theorem (see [OS] for definitions and details). Our results provide simple formulas for deducing, on minimal superreflexive Banach spaces (such as ℓp, 1< p < ∞), the existence of a convex function which is not oscillation stable from the existence of a Lipschitz function which is not oscillation stable. Let us fix some notation used in this paper. For a real Banach space X, we denote an equivalent norm on Xby k · k and by BXits closed unit ball under this norm. By convex function we will always mean continuous convex function. We will consider two fundamental topologies on the set of continuous functions defined on X:τκ(respectively τb) is the topology of uniform convergence on compact sets of X(resp. uniform convergence on bounded sets of X). The modulus of convexity of the norm k · k defined by δk·k(ε) = inf 1−    x+y 2    :x, y ∈BX;kx−yk ≥ ε(0 < ε < 2) is called of power type p(p≥2) if δk·k(ε)≥Kεp, for some K > 0. The concept of dyadic tree will play an important role in the second part of this work. Our trees are geometric trees contained in Xand defined as follows. The symbol αdenote a multi-index α= (α1 a α2 a...a αn)∈ {−1,1}<Nand |α|:= n. For n∈N, a dyadic (n, θ)-tree Tin Xis a set of the form xα∈X:α∈ {−1,1}<n satisfying the following two conditions: (1) xα=1 2xα a 1+1 2xα a -1, for all |α|< n. (2) kxα−xα′k ≥ θ > 0, for α6=α′. The point x∅will be called the root of the tree T. 1. The positive results Theorem 1. Let (X, k · k)be a Banach space. The norm k · k is locally uniformly convex (respectively uniformly convex) if and only if the following property holds: for every Lipschitz function fon X, the sequence of functions (fn)n∈Ndefined by the formula fn(x) := inf y∈Xnf(y) + n2kxk2+ 2kyk2− kx+yk2o(n∈N, x ∈X) is τκ-converging (resp. τb-converging) to f. Remark. For any function fon Xand n∈N, the function fndefined as above is a ∆-convex function. This follows immediately from the decomposition fn=cn−dn, with cn(x) := 2nkxk2and dn(x) := sup y∈Xnkx+yk2−2nkyk2−f(y)(x∈X). The functions cnand dnare clearly convex. 2 Proof of the Theorem 1. Let us see first that the property is necessary. So, let fbe a Lipschitz function on X. We have to show that the previously defined sequence (fn)n∈N(τKor τb)- converges to fif the norm k · k satisfies the corresponding rotundity condition. We begin with the following general result: Fact. For any point x∈X, (fn(x))n∈Nis an increasing sequence bounded above by f(x). This Fact follows immediately from taking y=xin the fn’s infimum formula and from the inequality 2kxk2+2kyk2−kx+yk2≥2kxk2+2kyk2−kxk+kyk2=kxk − kyk2≥0.(1) Since for K∈Nwe have that f(K·n)=Kf Kn, the previous Fact allows us to suppose without loss of generality that the Lipschitz constant of fis less than 1 (i.e. |f(x)−f(y)| ≤ kx−ykfor all x, y ∈X). We need to study of the infimum formula defining fnat a point x∈X. Thus, consider any point yfor which we have f(y) + n2kxk2+ 2kyk2− kx+yk2≤f(x).(2) As fis 1-Lipschitz, we deduce from (1) and (2) that nkxk − kyk2≤n2kxk2+ 2kyk2− kx+yk2≤f(x)−f(y)≤ kx−yk.(3) This last condition gives a relation between the norms of xand y. Actually, kyk can be controlled by kxkin the following way. Suppose that kyk ≥ 1 + kxk, then by (3) we get that 1≤kxk − kyk≤kxk − kyk2≤1 nkx−yk ≤ 1 nkxk+1 nkyk.(4) And we conclude for n≥3 that kyk ≤ n+1 n−1kxk ≤ 2kxk.Therefore, we see that if y satisfies (2) and n≥3 then kyk ≤ 2(1 + kxk).(5) Then, for n≥3 we deduce that fn(x) = inf kyk≤2(1+kxk)f(y) + 2nkxk2+ 2nkyk2−nkx+yk2.(6) In particular, the boundedness on bounded sets of fn(and, consequently, of dn= cn−fn) follows immediately. Moreover, the upper bound on kykgiven by (5) together with the condition (3) gives 0≤2kxk2+ 2kyk2− kx+yk2≤1 nkx−yk ≤ 1 nkxk+1 nkyk ≤ 3 n(1 + kxk).(7) The condition (7) gives the crucial step of the proof. In fact, if the norm k · k verifies one of the rotundity properties stated in theorem then by (7) ncan be 3 chosen big enough to necessarily enforce yto be close to x. Therefore, fn(x) must be close to f(x). Let us justify this assertion. Suppose that the sequence of functions fnis not compactly converging to f. Since the sequence of ∆-convex functions (fn)nis increasing and fis continuous, Dini’s theorem tells us that τκ-convergence of fnto fis equivalent to the pointwise convergence. Then, there exists a point x0∈Xsuch that (fn(x0))ndoes not converge to f(x0). As (fn(x0))nis increasing, there exists some ε0>0 such that for any n∈Nwe have fn(x0) + ε0< f(x0). By definition of fn, we can find a sequence of (yn)nin Xso that fn(x0) + ε0≤f(yn) + 2nkx0k2+ 2nkynk2−nkx0+ynk2+ε0≤f(x0).(8) Since fis 1-Lipschitz, we get from (8) that kx0−ynk ≥ f(x0)−f(yn)≥ε0>0.(9) Since ynverifies the condition (8), it follows from (7) that we have 0≤2kx0k2+ 2kynk2− kx0+ynk2≤3 n(1 + kx0k)−−−→ n→∞ 0.(10) But (9) and (10) show that the norm k · k can not be locally uniformly convex at x0(cf .Ch. II. Prop. 1.2 of [DGZ]). That proves the compact convergence of fnto fif the norm k · k is locally uniformly convex. A simple proof of the uniform convergence of the sequence fnin the uniformly convex case follows the same lines. We will give later a direct quantitative approach (see the proof of Theorem 3). If the sequence fndoes not converge uniformly on a bounded set of X, there is an ε0>0 and a bounded sequence (xn)nso that fn(xn) + ε0< f(xn), n∈N. Then, for each xn(n∈N) we can choose ynverifying f(yn) + 2nkxnk2+ 2nkynk2−nkxn+ynk2+ε0≤f(xn).(11) Similar reasonings as before imply from (11) that the next two statements are fulfiled: kxn−ynk ≥ f(xn)−f(yn)≥ε0>0 (12) 0≤2kxnk2+ 2kynk2− kxn+ynk2≤3 n(1 + kxnk)−−−→ n→∞ 0 (13) We have limn→∞ 3 n(1+kxnk) = 0 since the sequence (xn)nis bounded. Moreover, (5) implies that the sequence (yn) is also bounded. Therefore, (12) and (13) show that the norm k · k cannot be uniformly convex (cf. Ch. IV Lemma 1.5 [DGZ]). The necessity of the property is proved. Conversely, the property stated in the theorem is sufficient. We first show that the property of compact convergence implies the following claim. 4 Claim 1.1. For any sequence (xn)n∈N⊂Xand any point x0∈X, the condition limn→∞ 2kx0k2+ 2kxnk2− kx0+xnk2= 0 implies dx0,(xn)n= 0. Since the Claim 1.1 is also valid for any subsequence of the given sequence (xn)n, we can strengthen the conclusion of the claim to limn→∞ kxn−x0k= 0 and the locally convexity of the norm k · k is verified (cf .Ch. II. Prop. 1.2 of [DGZ]). Proof of the Claim 1.1. Consider the Lipschitz function g(x) = dx, (xn)n,x∈X. For x0∈Xand ε > 0, the pointwise convergence of the sequence (gn)nto gat x0 tells us that there exists Nε∈Nso that dx0,(xn)n=g(x0)≤inf y∈Xng(y) + Nε2kx0k2+ 2kyk2− kx0+yk2o+ε. (14) Using g(xn) = 0, we can evaluate the inequality (14) at y=xn(n≥Nε) and deduce that dx0,(xn)n≤Nεlim n→∞ 2kx0k2+ 2kxnk2− kx0+xnk2+ε=ε. Therefore, dx0,(xn)n= 0 and the claim is proved. If the property of uniform convergence on bounded sets of Xholds, the next claim, analogous to the previous one, also does. Claim 1.2. Let (xn)n∈Nand (yn)n∈Nbe two bounded sequences in Xsuch that limn→∞ 2kxnk2+ 2kynk2− kxn+ynk2= 0. Then limm→∞ dym,(xn)n= 0. Proof of the Claim 1.2. As before, consider the Lipschitz function d·,(xn)nand ε > 0. This time, the property of uniform convergence gives a positive integer Nε so that for all m∈Nthe next inequality is satisfied. dym,(xn)n≤inf z∈Xndz, (xn)n+Nε2kzk2+ 2kymk2− kz+ymk2o+ε. (15) Taking z=xmin the infimum of (15) we obtain dym,(xn)n≤Nε2kxmk2+ 2kymk2− kxm+ymk2+ε≤2ε, for mlarge enough, and the claim is proved. In order to finish with the proof, we shall show that the validity of Claim 1.2 implies that the norm k · k is uniformly convex. If not, (cf. Ch. IV Lemma 1.5 [DGZ]) there exists two bounded sequences (xn)n∈Nand (yn)n∈Nin Xsatisfying that lim n→∞ 2kxnk2+2kynk2−kxn+ynk2= 0 and kxn−ynk ≥ 1 (for all n∈N). (16) First, the norm k · k is locally uniformly convex (since Claim 1.1 clearly holds). It follows that the sequence (xn)nhas no norm cluster point. Indeed, if for some x0∈Xthere exists (xnk)k−→ kx0, then lim k→∞ 2kx0k2+ 2kynkk2− kx0+ynkk2= lim k→∞ 2kx0k2+ 2kxnkk2− kx0+xnkk2= 0 and therefore (ynk)k−→ kx0. A contradiction with the fact kxn−ynk ≥ 1, for all n∈N, of (16). Hence, passing to a subsequence, we can suppose that for some 1 > α > 0 we have that kxn−xmk ≥ α, for all n6=m. Now, we need the following technical lemma, whose proof will be given later. 5 Lemma 1.3. Let be (λn)n∈N⊂[0,1] and (xn)n∈N,(yn)n∈Ntwo bounded sequences in Xso that 2kxnk2+ 2kynk2− kxn+ynk2−−−→ n→∞ 0. Then the sequence of convex combinations zn=λnxn+ (1 −λn)yn(n∈N) satisfies that 2kxnk2+ 2kznk2− kxn+znk2−−−→ n→∞ 0. For any n∈N, take 0 ≤λn≤1 such that for zn:= λnxn+ (1 −λn)ynwe have kxn−znk=α 2. Then, the Lemma 1.3 and the Claim 1.2 used jointly imply that limm→∞ d(zm,(xn)) = 0. But, kxn−znk=α 2>0 and also for n6=m kxn−zmk ≥ kxn−xmk − kxm−zmk ≥ α−α 2=α 2>0, a contradiction.  Proof of the Lemma 1.3. By the inequality 0≤kxnk − kynk2≤2kxnk2+ 2kynk2− kxn+ynk2−−−→ n→∞ 0,(17) we have that lim n→∞ kxnk − kynk= 0. Since the sequences (xn)nand (yn)nare bounded we also have that lim n→∞ kxnk2− kynk2= 0.(18) But then we deduce from (17) and (18) that lim n→∞ kxnk2−   xn+yn 2   2= lim n→∞ kynk2−   xn+yn 2   2= 0.(19) On the other hand, using the convexity of the function k · k2we get the following general lower estimate for 0 ≤λ≤1 kλx+(1−λ)yk2≥    x+y 2    2 −1−2λmax (    x+y 2    2 −kxk2 ,    x+y 2    2 −kyk2). Putting (17),(18),(19) and the last inequality together, we obtain that 0≤2kxnk2+ 2kznk2− kxn+znk2 = 2kxnk2+ 2kλnxn+ (1 −λn)ynk2− k(1 + λn)xn+ (1 −λn)ynk2 ≤2kxnk2+ 2λnkxnk2+ 2(1 −λn)kynk2−4    1 + λn 2xn+1−λn 2yn    2 ≤2kxnk2+ 2kynk2− kxn+ynk2+λnkxnk2− kynk2 + 4λnmax (    xn+yn 2    2 − kxnk2 ,    xn+yn 2    2 − kynk2)−−−→ n→∞ 0. The lemma is proved and this concludes the proof of Theorem 1. Remark. The infimum formula used for the definition of (fn) in the Theorem 1 is closely related to the well-known inf-convolution formula of fby nk · k2: f(nk · k2)(x) := inf y∈Xf(y) + nkx−yk2(n∈N,x∈X).(20) 6 In fact, these two infimum formulas are identical if the norm k · k is a Hilbertian norm, because of the parallelogram identity. However, for a non-Hilbertian norm k · k the functions given by the inf-convolution formula can not be expressed in general as ∆-convex functions. As a corollary of the above remark, we have that the formula of Theorem 1 converges uniformly on Xfor case of a Hilbertian norm k · k (since it is well-known that the inf-convolution formula of (20) converges uniformly on the whole space X, see [LL]). The question that naturally arises is whether this remains true or not for a general uniformly convex norm. The answer is given in the following proposition. Proposition 2. Let (X, k · k)be a Banach space. If for every Lipschitz function f on Xthe sequence of functions fn(x) := inf y∈Xnf(y) + n2kxk2+ 2kyk2− kx+yk2o converges to funiformly on X, then the modulus of convexity of the norm k · k is of power type 2. Proof of the Proposition 2. This uniform convergence property has the following consequence analogous to Claim 1.1 and Claim 1.2 and whose proof is identical to that of Claim 1.2. Claim 2.1. If (xn)n∈Nand (yn)n∈Nare two sequences (not necessarily bounded) in Xsatisfying that limn→∞ 2kxnk2+ 2kynk2− kxn+ynk2= 0, then they also verify that the limm→∞ dym,(xn)n= 0. By Claim 1.2, any norm which satisfies the conclusion of Claim 2.1 is uniformly convex. In fact, Claim 2.1 insures that the modulus of convexity of the norm k · k is of power type 2. If not (see [H]), there exists two sequences (xn)n∈Nand (yn)n∈N such that 2kxnk2+ 2kynk2<kxn+ynk2+1 nkxn−ynk2xn6=yn,n∈N.(21) If we take un:= xn kxn−ynkand vn:= yn kxn−ynkin (21) we obtain that 0≤2kunk2+ 2kvnk2− kun+vnk2<1 n−−−→ n→∞ 0,(22) kun−vnk= 1.(23) Since the norm k·k is uniformly convex and conditions (22) and (23) holds, the sequences (un)nand (vn)ncan not be bounded. Notice that we have also from (22) that limn→∞ kunk − kvnk= 0. Thus, passing to a subsequence we can suppose that sup{kunk,kvnk} + 1 <min{kun+1k,kvn+1k} for all n∈N =⇒ kun−vmk ≥ kun|| − kvmk≥1 (n6=m). (24) Assembling (23) and (24) we deduce that dvm,(un)n= 1 (m∈N). But Claim 2.1 and (22) imply that limmdvm,(un)n= 0 which is a contradiction.  Theorem 3 below provides a converse to Proposition 2 and an explicit formula for uniform approximation of Lipschitz functions by ∆-convex functions on superreflexive Banach spaces. The validity of this formula depends upon the existence in the superreflexive space of an equivalent norm which is enough rotund. 7 Theorem 3. Let (X, k · k)be a Banach space whose norm k · k has its modulus of convexity of power type p(p≥2). Then for every Lipschitz function fon Xthe following sequence of ∆-convex functions fp n(x) = inf y∈Xnf(y) + n2p−1kxkp+ 2p−1kykp− kx+ykpo(n∈N, x ∈X) converges to funiformly on X. Proof of the Theorem 3. We essentially need the following lemma to prove the result. Lemma 3.1. If the modulus of convexity of the norm k·k is of power type p(p≥2) then there exists a positive constant Ck·k ≤1such that for every pair x, y ∈Xthe following inequality holds: Ck·kkx−ykp≤2p−1kxkp+ 2p−1kykp− kx+ykp. Proof of the Lemma 3.1. Under the assumption of the lemma (see [H]), there exists a positive constant C′ k·k ≤2 such that for every pair u, v ∈Xwe have that ku+vkp+ku−vkp≥2kukp+C′ k·kkvkp.(25) The lemma follows from the change of variables x=u+v 2and y=u−v 2in (25). Given a Lipschitz function fon X, we use the previous Lemma 3.1 and the fact that fp n(x)≤f(x) (for all x∈Xand n∈N) to obtain the following chain of inequalities: f(nCk·kk · kp)≤fp n≤f(n∈N).(26) Since p≥2, the previous inf-convolution formula in (26) converges to funiformly on X(see [LL]). Therefore, the same is true for (fp n)n. For instance, if fis a 1-Lipschitz function we get the following estimate kf−fp nk∞≤ kf−f(nCk·kk · kp)k∞≤1 nCk·k 1 p−1 . The first consequence that stems from the Theorem 1 is another proof of a result due to G.A. Edgar ([E]). Corollary 4. Let XBanach space with an equivalent locally uniformly convex norm. Then the σ-fields of Borel sets for the norm and weak topologies are the same. Proof of the Corollary 4. The first part of the Theorem 1 affirms that the existence of a locally uniformly convex norm on Ximplies the following: ∀f:X→RLipschitz ∃(cn)n,(dn)n⊂Conv(X) such that f=τκlim n→∞(cn−dn). Let us check that this property implies the equivalence of the two Borel families, Bor(X, k · k) and Bor(X, w). Obviously, Bor(X, w)⊆Bor(X, k · k). To see the other inclusion, take Fak · k-closed set of X. Then consider the Lipschitz function f(·) = dist(·, F). Note that the previous property implies that fis the pointwise limit of a sequence of w-Borel functions (because every continuous convex function is w-lower-semicontinuous). Therefore, fis w-Borel and so Fis w-Borel.  Now, we proceed to state two applications of Theorem 1 and Theorem 3 to the study of superreflexive Banach spaces. Actually, we will show in the next section that both of them characterize superreflexivity. 8 Corollary 5. Let Xbe a superreflexive Banach space. Denote by Conv(X)the set of continuous convex functions on X,Convb(X)the subset of Conv(X)consisting of the functions which are bounded on bounded sets and UCb(X)the class of functions on Xwhich are uniformly continuous on bounded sets of X. Then spanτbConvb(X)=UCb(X). Proof of the Corollary 5. From the Prop. 1.6 of [Ph] follows immediately that Convb(X) = Conv(X)∩ UCb(X)⊆ UCb(X). Let us show the τb-density of the ∆-convex functions in the set UCb(X). Notice that the set of Lipschitz functions on Xis dense in UCb(X) under the topology τb. This fact can be proved using again the inf-convolution formula. More precisely, given f∈ UCb(X) and bounded on X,f(nk · k)(x) is a sequence of Lipschitz functions τb-converging to f. Since the bounded functions of UCb(X) are clearly τb-dense in UCb(X), the density of the Lipschitz functions is therefore deduced. Thus, the corollary holds if we show that the convex functions {cn, dn}n∈Nobtained in the proof of Theorem 1 are in Convb(X). Clearly, cn(·) = 2nk · k2∈ Convb(X) and, as we remarked during the proof of this theorem, dn=cn−fnis also bounded on bounded sets of X. Corollary 6. Let Xbe a superreflexive Banach space. With the same notations of the previous corollary, one has spanτuConvb(X)⊃ UC(X) (where τuis the topology of uniform convergence on X). Proof of the Corollary 6. As we remarked during the proof of the previous Corollary 5, the result is proved if we show the τu-density of the subset of ∆-convex functions Convb(X) in the set of uniformly continuous functions UC(X). But this follows from Theorem 3 and Pisier’s renorming theorem ([P]) that gives an equivalent norm with modulus of convexity of power type p(for some p≥2) on every superreflexive Banach space. Remark. For a simpler and more geometrical proof of Pisier’s theorem we refer to [L]. 2. The negative results In this part, we will show that the rotundity conditions on the norm needed in Theorem 1 can not be dropped. For instance, even the pointwise convergence fails for some Banach spaces, as the following counter-example shows. Example 7. There exists a Lipschitz function on ℓ∞which can not be a pointwise limit of a sequence of ∆-convex functions. In the article [T], M. Talagrand proved that Bor(ℓ∞, w)&Bor(ℓ∞,k · k∞). Taking a k · k∞-closed, non w-Borel set Bthe function d(·, B) is a Lipschitz function which can not be the pointwise limit of any sequence of ∆-convex functions (since ∆-convex functions are w-Borel). On the other hand, the τb-density property of the span of Convb(X) in UCbis a characterization of superreflexivity. This conclusion comes from the following theorem. 9