f(xx + (1 - X)y) < Xf(x) + (1 - X)f(y) + e.

Transcription

1 PROCEEDINGS of the AMERICAN MATHEMATICAL SOCIETY Volume 118, Number 3, July 1993 A COUNTEREXAMPLE TO THE INFINITY VERSION OF THE HYERS AND ULAM STABILITY THEOREM EMANUELE CASINI AND PIER LUIGI PAPINI (Communicated by Andrew M. Bruckner) Abstract. Hyers and Ulam proved a stability result for convex functions, defined in a subset of W. Here we give an example showing that their result cannot be extended to those functions defined in infinite-dimensional normed spaces. Also, we give a positive result for a particular class of approximately convex functions, defined in a Banach space, whose norm satisfies the so-called convex approximation property. 1. Introduction In this paper we discuss the following problem: let A be a convex subset of a Banach space X. Consider an arbitrary e-convex function /: A >*H, that is, a function that satisfies for every x, y A and every X [0, 1] the inequality: f(xx + (1 - X)y) < Xf(x) + (1 - X)f(y) + e. Is it true that there exists a convex function g: A -* 9t such that \g(x)-f(x)\ < KeWx A, where K is a constant depending only on I? A positive answer was given by Hyers and Ulam [HU] (see also [Gr, C]) in the case X = 9\" (with any norm!); the best known estimates are with K = min(af, L ) where Mn = (n2 + 3n)/(An + 4) and L = m/2 for 2m~x < n < 2m (see also [Ge] for a discussion concerning these constants and related questions). We will show that in the infinite-dimensional case the stability theorem of [HU] does not hold. Also we shall give a positive answer when the space X and the functions satisfy some additional properties. For midpoint convex functions a counterexample is known [Ge]; for a positive result concerning //-approximately convex functions, which are also e-subadditive, see [K]. 2. Counterexample Let X be a Banach space. We denote by B(X) (resp. Br(X)) the unit ball (resp. ball of radius r) of X, i.e., B(X) = {x X: \\x\\ < 1} (resp. Received by the editors November 20, Mathematics Subject Classification. Primary 26B25; Secondary 46B99, 52A27. Key words and phrases. Approximately convex functions, Hyers and Ulam theorem, convex approximation property. During the preparation of the paper the authors were supported by grants from GNAFA-CNR and National Research Group "Functional Analysis" (M.U.R.S.T.) American Mathematical Society /93 $1.00+ $.25 per page

2 886 EMANUELE CASINI AND P. L. PAPINI Br(X) = {x X : \\x\\ < r}), and if A is a subset of X, by co.4 its convex hull. We say that the set A C X satisfies the condition C (e) if x, y A implies (x + y)/2 A + Be(X). We say that the set A satisfies the condition c (e) if x, y A and X [0, 1] implies Xx + (1 -X)y A + Be(X). Let lx be the Banach space of absolutely convergent sequences with the usual norm (that we will indicate by x i). We will denote by {e,} the standard basis of lx. Let C+ = {x lx : x, > 0}, the positive cone of lx, and B+ = C+nB(lx). For p (0, 1) set Sp = {x B+ : ^7-*? < 1} We need the following two propositions: the first one is in [L], but we give the proof since the paper is not very accessible, the second is in [CP]. Proposition 1. The set Sp satisfies condition C (2xlp~x - 1). Proof. Let x, y Sp such that ^=7 xf = t~ yf = 1 Then we have + r- x+y. ^ Xi + yt inf ^- - z = inf > r-z, inf f^l±zi _ z/). For positive numbers, sp + tp = zp implies 5 +? < z ; thus (2.2) inf g (*+*- )< inf g (f -«) + '(" -,,). * 4im\ I 1=1 Z '[=1 I I 1=1 Now let x and y be in Sp; define x', supports) as follows: y' (still in Sp and with disjoint *2n = 0> X2n_x = X, y2n-i ^> ^2«= J^i Then, by using (2.1) and (2.2), we obtain. x + y. x' + y' inf -z < inf -z, so we can suppose that x and y have disjoint supports. With this restriction, letting z, = (xi + yi)/2xlp we obtain Since x and y are arbitrary, Sp satisfies condition C (2x/p~x - 1). Proposition 2 (see [CP]). 7/" A satisfies C (e) ///en ^ satisfies c (2e). So, in particular, Sp satisfies condition c (21/p - 2). Now let us give the counterexample. For each p (0, 1), let fp: B+ c /'» 5H be defined in the following way: fp(x) = dist(x, Sp) = infz is/) \\x - z\\x. First of all we will prove that fp is a (2x/p - 2)-convex function. Take n > 0,

3 A COUNTEREXAMPLE TO THE HYERS AND ULAM STABILITY THEOREM 887 x, y B+, and zx, z2 Sp such that x - zi i < fp(x) + n and \y - z2 i < fp(y) + tl:> then, by using Proposition 2, we obtain fp(xx + (1 - X)y) = inf Ax + (l - X)y - z\\x zesp < inf(a jc-2,, + (l-a) y-z2 i + \\Xzi + (\-X)z2-z\\i) < Xfp(x) + (1 - X)fp(y) + n+inf \\Xzx + (1 - X)z2 - z, <Xfp(x) + (\-X)fp(y) + n + 2x'p-2. The conclusion follows from the arbitrariness of n. Now suppose that there exists K such that there exists a convex function g: B+ - SH satisfying /p(x) - #(x) < K(2X'P - 2). Take p (0, 1) such that #(2i/p_2)< 1/4. Let x = (\/n,..., \/n,0,0,...); we have (since fp(e{) = 0) J! i=i This is a contradiction, since an easy calculation shows that fp(x ) 1 - fli-'/p _ i as n» oo. Remark 1. The finite-dimensional version of this counterexample shows that, at least asymptotically, the constants Kn that appear in [C] are best possible. In fact, define 77 as the best constant such that for every convex subset A of 9*" and for every e-convex function / on A there exists a convex function g on A such that c?(x)-/(x) < 77 e for every x e A. Consider Sg = {x e W : Xi > 0, Y,"=x xf < 1}, and define, as before, fp(x) = infze5«x - z\\x. Then, if xn = (1/n,..., l/«), proceeding in the same way as before, we obtain /=i 1 - nx-x'p = fp(xn) < H (2X'» -2) + g(xn) This implies <Hn(2x'p-2) + Y *-" n i=\ <Hn(2^-2) + ±f^) i=i = 277 (21/"-2). l-nx~x/p log,// ^" ^ 4(2i/i>-i - 1)-4 as^r" + HfIP-2) Remark 2. Now consider the Hilbert space l2 = {{x } : 5Zn^i x2 < +00} with the usual norm. Then the previous example works, also, if one thinks of B+ as a subset of l2. Notice that the convex set B+ is a convex subset of l2 (since algebraically lx c l2) and, also, it is a closed subset in l2. Take, in fact, a

4 888 EMANUELE CASINI AND P. L. PAPINI sequence {x } c B+ such that x > x in the l2 norm. Then {x } converges to x in the weak-topology of l2 and so it converges coordinatewise. But, also, as a bounded sequence in /[ it has a subnet converging in the weak*-topology of lx (and in particular coordinatewise) to an element of lx. So x belongs to lx, thus to B+. Remark 3. As the previous remark pointed out, the stability problem is (in some sense) independent of the norm-topology of the Banach space in which the domain of our e-convex functions lie. A way to relate the norm to the functions is to ask that they must satisfy some lipschitz condition. Notice that the functions fp considered in our counterexample are 1 -lip, since it follows easily, from the definition of distance, that \fp(x) - fp(y)\ < \\x - y\\x. The same condition is not satisfied if we think of our example embedded in l2. In fact, as a simple calculation shows, for any fixed h our functions are not //-lip, for every p, in the /2-norm. This will follow directly from Theorem 1 of the next section. 3. A POSITIVE RESULT In this section, we prove that the construction of our counterexample is possible since lx lacks a convexity property called the "convex approximation property" (C.A.P. for short). We will say that a Banach space X has C.A.P. if for every e > 0 and r > 0, there exists an integer p = p(e,r) such that for every A c Br(X) we have coa c copa + Be(X), where cop A = {x X : x J2P=o a'xi' Xj X, a, > 0, 2T/L0 ai = 1} In other words, each element of co A can be e-approximated by a convex combination of no more than p vectors of A. We will say that X is B-convex [P] if there exist constants c > 0, p > 1 such that for every n and all independent random variables gx,..., g with values in X we have (3-D *(l> )<0J2E\\gi\rx. \ (=1 X' '=1 It is easy to construct a sequence of independent random variables with values in lx such that (3.1) does not hold. This implies that lx is not 77-convex and so it does not have C.A.P. since we have the following result of Bruck [B]. Proposition 3. A Banach space X has C.A.P. if and only If it is B-convex. Moreover, in this case, there exist constants c > 0, q > 1 depending on X and r such that, for every A C Br(X), we can choose p (in the definition of C.A.P.) via p < ce«/(1-«'. The main result is the following: Theorem 1. Let X a B-convex Banach space, h > 0, and A a bounded convex subset of X. Then, for every f: A 9\, which is e-convex and h-lip, there exists (for e sufficiently small) a constant K (depending on X, h, and diam(a)) such that there exists a convex function g: A 91 satisfying \f(x) -g(x)\ < Ke lg2 e. We will use the following lemma, the proof of which can be found in several papers and in [C] with the best known constants.

5 A COUNTEREXAMPLE TO THE HYERS AND ULAM STABILITY THEOREM 889 Lemma 1. Let X be a B-convex Banach space and f: A c X > 9t an e-convex function. Then for xo,..., xp e A, an,..., ap e [0, 1], ax H-hap = 1, and all integers p, we have f(y?i=o aixi) < X^=o aif(xi) + ^Kpe. (For the definition of Kn see the introduction.) Proof of Theorem 1. Suppose that A C Br(X) and let d = diam(a). We take Z = X R with z = (x, y)\\ = x + \y\ (x X, y R). It is easy to show that Z is a Ti-convex Banach space since X is 5-convex, and so it has C.A.P. Since f is //-lip then / is bounded on A. Let m = infxea/(x), M = supx6a f(x), and 770 = {z e Z : z = (x, y) > xea, f(x) <y< M}. We have to show that 770 is a bounded subset of Z. In fact, if we take zx, z2 770 we have z, - z2\\ = xi - x2 + \yx-y2\<d + M-m; but as a consequence of the lipschitz condition on / we have M - m < hd, so diam(770) < (1 + h)d. Take n > 0. Then there exists an integer p such that if z 6 co 770 there exist zq,..., zp Ho and ao,..., ap so that Then by using Lemma 1 v p p z - Y^ <*izi = x - Y a'x> + y - zz aiyi -' j=0 j=0 j=0 fx)=f(x) -f(iz a'x) + / (e a'x) < f(x)-f[y/alxl\ +Y,<*if(Xi) + 2Kpe p \ i=0 / j=0 p < h * - Ea'x> + Ea'y' ~y +y + 2Kp j=0 1=0 <h'n + y + 2Kpe (//' = max(//, 1)). Taking into account the upper bound for p given in Proposition 3 and the value of the constant K, we obtain (for p sufficiently large, that is, for n small) f(x) <h'n + ( 1 + lg2cni'^-o^e + y. Since n was arbitrary, we can choose: n = qe/(\ - q)h' \g2e (this is the minimum point of the function g(n) = h'n + (1 + lg2 cnql{x~q"l)e). Then we obtain (3.2) /(x)<27c: lg2e + y for some negative constant K. Now define, for x 6 A, gom = inf{y R : (x, y) co Ho}. Then by (3.2) we have f(x) - 2Ke\g2e < go(x) < f(x). It is not hard to prove that o is a convex function, and if we put g(x) = go(x) + Ke lg2 e we obtain which concludes the proof. f(x) - Ke lg2 e < g(x) < f(x) + Ke lg2,

6 890 emanuele casini and p. l. papini [B] [CP] References R. E. Bruck, On the convex approximation property and the asymptotic behavior of nonlinear contractions in Banach spaces, Israel J. Math. 38 (1981), E. Casini and P. L. Papini, Almost convex sets and best approximation, Ricerche Mat. (to appear). [C] P. W. Cholewa, Remarks on the stability of functional equations, Aequationes Math. 27 (1984), [Ge] R. Ger, Almost approximately convex functions, Math. Slovaca 38 (1988), 61-78; Errata: 423. [Gr] J. W. Green, Approximately convex functions, Duke Math. J. 19 (1952), [HU] D. H. Hyers and S. M. Ulam, Approximately convex functions, Proc. Amer. Math. Soc. 3 (1952), [K] Z. Kominek, On approximately subadditive functions, Demonstratio Math. 23 (1990), [L] J.-O. Larsson, Studies on the geometrical theory of Banach spaces: part 5: Almost convex sets in Banach spaces of type p, p > 1, Ph.D. Thesis, Uppsala Univ., [P] G. Pisier, Sur les espaces de Banach qui ne contiennent pas uniformement de l\, C. R. Acad. Sci. Paris Ser. A 277 (1973), DlPARTIMENTO DI MATEMATICA, UnIVERSITA DI BOLOGNA, PIAZZA PORTA S. DONATO, 5, I Bologna, Italy address, P. L. Papini: PAPINI@DM.UNIBO.IT