Preiss, D.

Zajíček, L.

Institute of Mathematics AS CR, v. v. i.

WSAA 11David Preiss; Luděk ZajíčekStronger estimates of smallness of sets of Frechet nondifferentiability of convexfunctionsIn: Zdeněk Frolík (ed.): Proceedings of the 11th Winter School on Abstract Analysis. CircoloMatematico di Palermo, Palermo, 1984. Rendiconti del Circolo Matematico di Palermo, Serie II,Supplemento No. 3. pp. [219]--223.Persistent URL: http://dml.cz/dmlcz/701314Terms of use:© Circolo Matematico di Palermo, 1984Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access todigitized documents strictly for personal use. Each copy of any part of this document must containthese Terms of use.This paper has been digitized, optimized for electronic delivery and stampedwith digital signature within the project DML-CZ: The Czech Digital MathematicsLibrary http://project.dml.czSTRONGER ESTIMATES OF SMALLNESS OF SETS OF FRECHET NONDIFFERENTI-ABILITY OF CONVEX FUNCTIONS D*Preiss,L.Za;jiSek A well-known result of Asplund £1J says that every continuous convex function on a real Banach space X with a separable dual is Frechet differentiable at every point of some residual subset of X . This result was generalized by Robert £53 , who proved that every monotone operator on such a space X is single-valued and upper-semicontinuous at every point of some residual subset of X . The problem of finding more restrictive notions of small-ness of the set of points of Frechet nondifferentiability of con-tinuous convex functions on X was partially solved in £4J • There it was shown that this set is even (^-porous, but that it need not be a null set for any given Radon measure. Here we impro-ve the results of £4J in two ways. Firstly, we introduce the notion of angle smallness of a subset of X , which is stronger than (j -porosity and we prove that for every monotone operator on a Banach space with a separable dual is single-valued and upper semicontinuous at all points of X except those which belong to some angle small set. Secondly, we introduce the notion of ball smallness and we show that for every ball small set in a Hilbert space there is a convex continuous function which is not Frechet differentiable at all points of this set. Although the notions of angle smallness and of ball smallness do not coincide, they seem to be rather close together and hence they give in Hilbert spaces rather good estimates of the magnitude of the sets of points of Frechet nondifferentiability of continuous convex fun-ctions. Let X be a real Banach space. The open ball with the center x £ X and the radius r > 0 is denoted by B(x,r) . A set M C X is said to be oC -angle porous ( where ot > o ) , if for every x£M and every £>0 one may find z e B(xf£) and f e f such that 220 D.PREISS,L.ZAJfCEK M A { y e X ; < y - z , f > > Klff(| l |y-z| l i - * • If, for every c{ p o s i t i v e , M can be wr i t t en as a countable union of 0( -angle porous s e t s , we say tha t M i s angle small . A se t MCX i s ca l led r - b a i l porous (where r > 0 ) i f for every x£ M and every £ > 0 there i s u6X such tha t |[u-x| | » r and M C\ B(u,r-£) = & . If M can be wr i t t en as a union of a countable family of sets, each of which is r-ball porous for some r > 0 , we say that M is a ball small set. It is easily seen that every ball small set in a separable Hilbert space is angle small and that, if a set M can be written as a union of a countable family of sets each of which is oC -angle porous for every o<> 0 , then M is a ball small set. This shows that, in separable Hilbert spaces, the notions of angle smallness and ball smallness are rather close. Theorem 1. Let X be a real Banach space with a separable "" • % dual and let T:X->X be a monotone possibly multivalued operator with an arbitrary domain D(T) =£x; T(x) 4 Jtfj . Then there is an angle small set AcD(T) such that T is single-valued and upper-semicontinuous at every x6D(T) - A . Proof. We need to show that the set A = C xeD(T) ; lim diam T(B(x,<F)) > 0 J is angle small . Let A_ = (xeD(T) ; j""-i> 0 . 4 . A _ n lim diam T(B(x,<F)) > n" j and let C be a countable dense subset of X • Whenever o< > 0 is given, we decompose A = U A„ -J* -i "PfcC* • where A n ^ -= (xfcAR ; dist(f,T(x)) < */2nJ: . Clearly, it suffi-ces to prove that each of the sets A « is rX -angle porous. Let n,f, x£ A^ « and r > 0 be fixed. There is zeX , /(z-x[( < r II, i. such that //Tz-fJ > 1/2n for|> some T2€T(z) . To show that An f is c* -angle porous it is sufficient to prove that An',f n £ y e X ; <y-z>Tz-f> > o(//Tz-f// // y-z/Q -- X . Whenever ye*D(T) and <y-z,Tz-f> > o< || T2-f// //y-z// , and whenever Ty£T(y) we have <y-z,Ty-f> = <y-z,Ty-T2> + <y-z,Tz-f> £ <y-z,Tz-f> > > oCl|T2-f II || y-z|| ̂  0(||y-z [| / 2n . Hence ff Ty-f// > (*/2n for every T eT(y) which, according to the definition of the set An f , shows that y ̂  An f . Remark. Using the Gregorys method of separable reduction (cf. C23 , p.Hl) , one can recover the result of Kenderov [jj , according to which every monotone non-empty valued operator T:X->X on an Asplund space X is single-valued and upper-semi-continuous at points of some residual subset of X . Corollary 1. Let f be a continuous convex function on a real Banach space with a separable dual. Then the set of points STRONGER ESTIMATES OF SMALLNESS OF SETS 221 at which f is not Frechet differentiate is angle small. Theorem 2. Whenever M is a ball small set in a Hilbert spa-ce X' t there is a continuous convex function f on X which is not Frechet differentiable at any point of M. Proof. We assume first that M is a bounded r-ball porous set. Let N « M U (X-B(OtR)) where R>2r is chosen so that (1) M c B(0tR-2r) . Let C be the closed convex hull of the set «£(ytt)gNxR ; t > /Jy/|2j. Then the function f(x) « inf -ft £R ; (xtt) £ Cj is easily seen to be well-defined, convex and continuous on X . It is also clear that (2) f ( x ) > / / x | / 2 for x£X and (3) f(x) zz //x|/ 2 for x^M . We prove that f is not Frechet differentiable at any point of M. Whenever X£ M t we first note that the derivative of ||y || at x equals 2x t hence (2) and (3) imply 2xe^f(x) . On the other hand, for every £ > 0 there is ufcX such that l/u-x // = r and B ( utr\(T"--t) O M -= j2T . Tor z £ B ( utr\Tn-"£) we have i /zf j 2 * Jfx/f2 + 2 ( x , z - x ) - Bu-xJJ 2 + tfz-u/f2 + 2 ( u - x t z - x ) i .> |/x f| c + 2 ( x t z - x ) - £ r + 2 ( u - x t z - x ) whicht together with Bfu-ryTrjT) O N « Jgf , shows tha t f ( z) > J/x J/2 + 2 ( x , t - x ) - £ r 2 + 2 ( u-x tz-x^ for every z eX. For z * x + £ ( u - x ) we get f(z) - f ( x ) ^ 2 ( x t z - x ) + £ r2 « 2 ( x , z - x ) + r 2 / ( z - x / / , which shows tha t f i s not Frechet d i f f e r en t i ab l e at x . To prove the general case we decompose M » \J Mn , where each Mn i s a bounded r - ba l l porous set and we construct the functions f by the above method. Since 0 £ f n ( x ) i An + // x |/ for some An t i t suff ices to put f = S cw f t where c are su f f i c i en t ly 47 mmmm n n n small positive numbers. Remarks. 1. If a Lipschitz function g:R->R is not differe-ntiate at every point of some residual subset of R then its graph serves as an example of a ^-porous set which is not angle small. Hence Corollary'1 gives a stronger result than Theorem 1 from £4j . 2. One can find a J*-convex surface in Rn which is not a ball small set. Hence ( cf. t6j) one can find a convex fun-ction in Rn such that its points of nondifferentiability do not form a ball small set. Therefore ball smallness is not a characte-rization of the magnitude of sets of points of Frechet nondifferen-222 ' D.PREISS,L.ZAJ1CEK tiability even in finite dimensional Hilbert spaces. 1 n 3. Any C -smooth surface in R which is not a countable union of a -convex surfaces gives an example of an angle small set which is not container! in the set of points of nondiffe-rentiability of any convex function on Rn . Hence angle smariness is also not a characterization of the magnitude of sets of points where a continuous convex function is not Frechet differentiable. 4* In one (very special!) case our notions give even an exact characterization of sets of points of nondifferenti-ability of convex functions. Namely, in a one-dimensional Banach space angle small as well as ball small sets are exactly counta-ble ones. There is quite a number of open problems in this area. Among them the following one seemed to us most intriguing. To motivate it , let us note that for any closed convex set C with empty in-terior the function x ^ dist(x,C) is Frechet nondifferentiable at every point of C , and that the sets of points of Gateaux nondifferentiability of any continuous convex function on a separa-ble Banach space can be covered by countably many J"-convex surfa-ces. (This is even a characterization of their magnitude, see[63). Problem 1. Can the set of points of Frechet nondifferentiabili-ty of a continuous convex function on a Banach space with a separa-ble, dual or on a separable Hilbert space be covered by countably many closed convex sets with empty interior and by countably many J -convex surfaces? Another very natural problem is, whether it is possible to re-cognize good differentiability properties of continuous convex fun-ctions on a Banach space using the given norm only. Problem 2. Is there a notion of smallness of sets in separable Banach spaces with the following two properties? a) If X is separable, then the norm on X is Frechet diffe-rentiable everywhere except points of some small subset of X. b) If X is separable and its norm is Frechet differentiable everywhere except at points of some small subset of X , then X is separable. In particular, is b) true with "small" meaning "angle small" or " (f -porous" ? STRONGER ESTIMATES OF SMALLNESS OF SETS 223 REFERENCES 1. ASPLUND E. "Frechet differentiability of convex functions", Acta Math. }2± (1968), 31-47. 2. GILES J.R. "On the characterization of Asplund spaces",J.Austral. Math.Soc.(Series A)32(1982), 134-144. 3. KENDEROV P.S. "Monotone operators in Asplund spaces", C.R.Acad. Bulgare Sci. 30(1977), 963-964. 4. PREISS D., ZAJiCEK L. "A note on the Frechet differentiation of convex functions in a Banach space with a separable dual", to appear. 5. ROBERT R. "Une generalisation aux operateurs monotones des the-oremes de differentiabilite d'Asplund", Lecture Notes in Economics and Math. Systems, vol. 102, Springer-Verlag 1974, 168-178. 6. ZAJiCEK L. "On the differentiation of convex functions in finite and infinite dimensional spaces", Czechoslovak Math.J. 29(104)(1979), 340-348. CHAIR OF MATHEMATICAL ANALYSIS, FACULTY OF MATHEMATICS AND PHYSICS, CHARLES UNIVERSITY, S0K0L0VSK.A 83, 186 00 PRAGUE, CZECHOSLOVAKIA 

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Stronger estimates of smallness of sets of Frechet nondifferentiability of convex functions

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Stronger estimates of smallness of sets of Frechet nondifferentiability of convex functions

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