ON DC FUNCTIONS AND MAPPINGS

ON D.C. FUNCTIONS AND MAPPINGS JAKUB DUDA, LIBOR VESELY , AND LUDE K ZAJIC EK Abstract. A function on a Banach space is called d.c. if it is a dierence

of two continuous convex functions. A theory of d.c. mappings between Banach spaces (which generalize P. Hartman's notion of d.c. mappings between Euclidean spaces) was built in [31]. In the introduction we present basic information on d.c. functions and mappings and brie y comment on several recent articles on this topic. In Section 2 we consider the question whether each C 1 1 function (or mapping) de ned on an open convex subset A of a Banach space is d.c. As a consequence of our results we obtain a negative answer to a problem from [31]. In the Section 3 we consider the question whether a d.c. mapping de ned on an open convex subset C of a Banach space X is Lipschitz if it has a Lipschitz control function. We show that the answer is negative in general but it is positive if C is bounded or if C contains (a translate of) a nonempty open convex cone. In Sections 4 and 5 we present several observations which concern natural questions about d.c. mappings between Banach spaces. ;

1. Introduction In this section we present basic information on d.c. functions and mappings. All the facts needed in the sequel will be formulated precisely. Further, we brie y comment on some articles which appeared after the survey article [14] on d.c. functions and [31] (where a theory of d.c. mappings between Banach spaces was built). In particular, we indicate which of the ten problems formulated in [31] were already solved. At the end of this section we brie y describe new results of the present paper. For readers who are not interested in d.c. mappings, we formulate explicitely what our results (of Sections 2 and 3) say for d.c. functions. 1.1. Basic results on d.c. functions and mappings. A real function f de ned on R n (or, more generally, on a convex subset of R n ) is called a d.c. function if it is a dierence of two convex functions. Such functions are sometimes labeled as -convex, -convex (or delta-convex) functions. We prefer here the name \d.c. function" mainly because the term \-convex function" 1991 Mathematics Subject Classi cation. Primary: 47H99; Secondary: 26B25. Key words and phrases. d.c. function, d.c. mapping, normed linear space, Lipschitz condition. . 1

2

JAKUB DUDA, LIBOR VESELY , AND LUDE K ZAJIC EK

is also used for functions, which are \almost convex" with an error at most

> 0.

D.c. functions of one real variable, considered by numerous mathematicians (at least starting from 1911), have a very simple internal characterization: they are precisely inde nite integrals of functions with locally bounded variation. The rst who considered d.c. functions of several variables was probably A. D. Alexandrov [1] (cf. also [2]) in 1949, motivated by the geometry. A survey on d.c. functions on R n is contained in [14]. Note that such functions were used in many articles on the non-smooth optimization theory (cf. e.g. [14], [25]). In contrast to the case n = 1, it seems that no simple and usefull internal characterization of d.c. functions on R n (n > 1) is known. The class of d.c. functions on R n contains all C 2 functions and is stable under many operations. In fact, it is not only a linear space (generated by the convex functions) but also an algebra and a lattice. These stability facts were generalized in an important article [13] by P. Hartman in 1959. He de ned d.c. mappings between Euclidean spaces as mappings with d.c. components and proved that the composition of two d.c. mappings is d.c. He also proved that a function which is locally d.c. on an open convex subset C of R n is (globally) d.c. on C . The notion of d.c. functions on in nite-dimensional spaces arises quite naturally in the investigation of Gâteaux dierentiability of convex functions [32] and in the abstract approximation theory [33]. Since only continuous convex functions on in nite-dimensional spaces are interesting, a function de ned on an open convex subset of a normed linear space is called d.c., if it is a dierence of two continuous convex functions. In [31], d.c. mappings between normed linear spaces were studied. Their following de nition is not \canonical" and it is not a priori obvious that it gives the \right" generalization of (the canonical) Hartman's notion of a d.c. mapping between Euclidean spaces. De nition. Let X and Y be normed linear spaces, C X be an open convex set and F : C ! Y be a mapping. We say that F is a d.c. mapping if there exists a continuous convex function f on C such that y F + f is a continuous convex function for each y 2 Y ; kyk = 1. Every such f is called a control function for F . Other natural de nitions are the following: (i) We say that F is a weakly d.c. mapping if y F is a d.c. function on C for each y 2 Y . (ii) Let Y be an ordered normed linear space. We say that F is an order d.c. mapping if F is a dierence of two continuous convex (in the sense of the order on Y ) operators on C .

ON D.C. FUNCTIONS AND MAPPINGS

3

It is easy to see that all the three notions (d.c., weakly d.c., order d.c.) coincide in the case Y = R n . Observe that a mapping F is d.c. i it is in a sense \uniformly weakly d.c". Indeed, F is d.c. with a (continuous convex) control function f i, for each y 2 Y ; kyk = 1, we have y F = gy ? f with a continuous convex function gy . An alternative equivalent de nition of d.c. mappings, possibly more natural than the de nition above, is based on the following fact (see [31] , pp. 11,14) that, roughly speaking, f controls F i a \measure of non-anity" of F is smaller than the same \measure of non-anity" of f . Theorem 1. Let X; Y be normed linear spaces, A X an open convex set and let continuous F : A ! Y; f : A ! R be given. Then the following conditions are equivalent. (i) F is d.c. with a control function f . (ii) kF (x) + F (y) ? F (x + y)k f (x) + f (y) ? f (x + y) whenever x 2 A; y 2 A; 0; 0; + = 1. (iii) k F (x)+2 F (y) ? F ( x+2 y )k f (x)+2 f (y) ? f ( x+2 y ) whenever x; y 2 A. We believe that the de nition of d.c. mappings adopted in [31] (together with its equivalent formulation from Theorem 1) is the most natural one because of its nice properties which fail to hold for the other two de nitions. For example, composition theorems for weakly d.c. or order d.c. mappings do not hold (see Section 4) while, for locally d.c. mappings, the following generalization ([31], p. 27) of Hartman's composition theorem holds. Theorem 2. Let X; Y; Z be normed linear spaces and let A X; B Y be open sets. Let a mapping F : A ! B be locally d.c. on A and let G : B ! Z be locally d.c. on B . Then the composite mapping G F is locally d.c. on A. Further, the known characterizations of real d.c. functions of one real variable have natural extensions [31] to the case of d.c. mappings from the real line into a normed linear space (Theorem 3). For one of them we need the following notion of convexity which goes back to de la Vallee Poussin (1908; cf. [20]). De nition. Let X be a normed linear space and f : [a; b] ! X be a mapping. For every partition D = fa = x0 < x1 < < xn = bg of [a; b] we put

n?1 X

f (xi+1 ) ? f (xi )

f ( x ) ? f ( x ) i i ? 1 b

: K (f; D) = ? a

i=1

xi+1 ? xi

xi ? xi?1

The convexity of f on [a; b] we de ne as Kab f = sup Kab (f; D); where the supremum is taken over all partitions D of [a; b].

4


Theorem 3. Let X be a Banach space and f : (a; b) ! X be a continuous

mapping. Then the following conditions are equivalent. (i) f is d.c. on (a; b). (ii) f+0 (x) exists for each x 2 (a; b) and f+0 has locally nite variation on I . (iii) Kcd f < 1 for each interval [c; d] (a; b). D.c. mappings also copy many smoothness properties of a (continuous convex) control function f . In particular, the following results [31] hold. Theorem 4. a) Every d.c. mapping is locally Lipschitz. b) Every d.c. mapping has all one-sided directional derivatives at all points. c) Let F be a d.c. mapping with a control function f . Then: (i) If f is Frechet dierentiable at a point x then F is Frechet (even strictly) dierentiable at x. (ii) If f is Gâteaux dierentiable at a point x then F is Gâteaux dierentiable at x. d) Every d.c. mapping on an open convex subset of an Asplund or weak Asplund space is generically Frechet (even strictly) dierentiable or Gâteaux dierentiable, respectively. The applications of d.c. mappings between in nite-dimensional spaces are not too rich for the present; we know only about the following ones: (i) The notion of a d.c. mapping was used already in [28] for a natural de nition of n-dimensional d.c. surfaces in in nite dimensional spaces. Such surfaces are natural for description of some sets of singular points of convex functions and monotone operators. (ii) Some mappings which naturally arise in the theory of integral and dierential equations (e.g. Nemyckii's and Hammerstein's operators) are d.c. under some (not too sever) conditions and consequently have many good dierentiability properties (cf. [31]). (iii) The composition theorem for d.c. mappings was used by Cepedello-Boiso to prove a theorem ([6], Theorem 1) on \interpolation" by d.c. functions on Banach spaces. (iv) The same composition theorem was used in [34] to prove a result concerning (relative) dierentiability properties of convex functions on Banach spaces. We believe, however, that the notion of d.c. mappings is so natural that it will nd further applications.

1.2. Comments on some resent results. At the end of [31], ten problems on d.c. functions and mappings were formulated. Four of them were solved (by counterexamples) in [15] and [9]. A further problem is solved in the present article. Let us describe these results.


5

Problem 5 of [31] asked whether each locally d.c. mapping F : A ! Y (where X; Y are Banach spaces and A X is an open convex set) is d.c. on A.

Already Hartman [13] mentioned that the proof of his theorem (which says that the answer is positive for nite-dimensional X; Y ) works for functions on nite-dimensional spaces only. However, it is not easy to nd a counterexample showing that the answer to Problem 5 is negative. As far as we know, it was rst published in [15], where the authors construct a counterexample in `2 (they note without a proof that a similar example can be constructed in each in nite-dimensional space). Moreover, they construct a real function on `2 which is d.c. on each bounded open subset but is not d.c. On the other hand (as mentioned in [31], p. 14), Hartman's proof can be easily extended to the case dim X < 1; dim Y = 1. Note also that (rather weak) valid versions with dim X = 1 of Hartman's theorem are proved in [17] and [15]. To motivate Problem 3 of [31] consider a d.c. function f de ned on an open convex subset A of a Banach space X , which is Frechet dierentiable at a point a 2 A. It is natural to ask whether f admits a decomposition f = g ? h, in which both continuous convex functions g; h are Frechet dierentiable at a. This question has a negative answer already in the case X = R 2 . Indeed, it is not dicult (cf. [22] and [31], p. 36) to nd a d.c. function f which is dierentiable at a but it is not strictly dierentiable at a and use the fact that for a convex function Frechet dierentiability implies strict dierentiability. Thus it is natural to ask, whether f admits such a decomposition if it is strictly dierentiable at a. Also this question has a negative answer; the authors of [15] constructed a counterexample in which f is a function on R 2 . It provides a negative answer to Problem 3, which asked whether a d.c. mapping which is strictly dierentiable at a has a control function which is strictly dierentiable at a. (An another counterexample to Problem 3, which is given by a d.c. mapping from `1 to `1, can be found in [9].) Problem 1 of [31] asked whether F ?1 is locally d.c. on B , whenever X; Y are Banach spaces, A X; B Y are nonempty open convex sets and F : A ! B is a bijective locally d.c. mapping such that F ?1 is locally Lipschitz on B . It was proved in [31] that this problem has the positive answer if X; Y are nite-dimensional and it has an \almost positive" answer if X is an Asplund space: then there exists an open set B~ B dense in B such that F ?1 is locally d.c. on B~ . In [9], Problem 1 was solved in negative by a counterexample with X = Y = 1 ` . Moreover, the construction in [9] gives also a negative answer to Problem 2 of [31] (F has an invertible strict derivative at a point a 2 A and still (F=U )?1 is locally d.c. on F (U ) for no open neighbourhood U of a). The case X = Y = `2 remains open. Note also that (see [9]) no counterexample F to Problem 1 can be obtained in the form F = G + H , where G is a

6


\bi-locally d.c mapping" and H is a mapping with a nite-dimensional range.

Problem 6 of [31] reads as follows: Let X; Y be Banach spaces, A X be an open convex set and F : A ! Y be a mapping. Suppose that F ' is d.c. on (0; 1) whenever ' : (0; 1) ! A is d.c. Is then F locally d.c. on A ? This is only one possible version of problems which arise from the following natural rough question: Is it possible to characterize d.c. functions (or even mappings) of more variables in the language of \curves" (i.e. mappings of one real variable) only? Already A. D. Alexandrov [1] asked, whether a function on R 2 which is (rougly speaking) d.c. over each line must be d.c. It is essentially a version of Problem 6, in which X = R 2 , Y = R and only ane ' are considered. This version has a negative answer. This fact is stated in the survey article [14] (where it is attributed to Y. Yomdin); a simple counterexample can be found in [31], p. 35. In the present article we solve Problem 6 in negative (see the end of Section 2). Namely, there exists a mapping F : `3 ! `1 which is continuous and quadratic (and therefore, as we prove, satis es the assumptions of Problem 6) but it is not locally d.c. It is still possible that the following Problem 7 of [31], which is a \quantitative version" of Problem 6, has a positive answer. Let X; Y be Banach spaces, A X be an open convex set and F : A ! Y be a Lipschitz mapping. Suppose that there are a 0; b 0 such that (1.1) K01 F ' a K01 ' + b Lip(') whenever ' : [0; 1] ! A is Lipschitz. Is then F d.c. on A? It is possible to prove (Proposition 2.22(b) in [30]) that, if F is d.c. on A, then the corresponding a; b always exist locally. More precisely, every x0 2 A is contained in an open convex set A0 A such that, for some a; b 0, (1.1) holds for each Lipschitz curve ' : [0; 1] ! A0. A similar question is considered in [19]. This article contains a theorem which characterizes positively homogenous d.c. functions on R 3 (and thus it is very close to a characterization of d.c. functions on R 2 ) by the way rather similar to the way suggested in Problem 7. However, the geometrical construction of polyhedral surfaces needed in the proof is sketched only. Moreover, it seems that it contains a gap; at least it is not clear and thus validity of the theorem is not guaranteed. Finally, note that the Problems 4,7,8,9,10 of [31] are, as far as we know, still open. Besides already mentioned articles [15], [17], [6], [9], [19], let us mention several other recent articles on d.c. functions and mappings.


7

Articles [23], [24], [4], [5], [6] contain results on uniform approximation of Lipschitz functions by d.c. functions on (superre exive) Banach spaces. The subsets of a Banach space which are of the form fx : f (x) = 0g, where f is d.c., were considered in [26], [26], [24], [5] and [6]. The article [36] generalizes and makes more precise results of [35] concerning the recognizing of d.c. functions by an inductive process which uses the operation of a convex envelope of a function. The spaces of d.c. functions and mappings were recently considered in [27] and [8], respectively. D.c. mappings between normed linear spaces were considered further by R. Ger. In [10] he considered also more general mappings which are (possibly discontinuous) solutions of the inequality

F (x) + F (y )

f ( x ) + f ( y ) x + y x + y

? F 2

?f 2 :

2 2 In this case f need not be even convex, but it is always Jensen{convex. In [11] he introduced and investigated (Jensen) delta{convex mappings of n-th order (n 2 N ). 1.3. New results of the present article. In Section 2 we consider the question whether each C 1;1 function (or mapping) de ned on an open convex subset A of a normed linear space X is d.c. Recall that F : A ! Y is a C 1;1 mapping if its Frechet derivative F 0(x) exists at each point x 2 A and the mapping F 0 is Lipschitz on A. A positive answer is contained in [1] for functions on R n , and in [31] for mappings F : A ! Y , where X is a Hilbert space and Y an arbitrary normed linear space. Here we prove that the answer is positive if X admits an equivalent norm whose modulus of convexity is of power type 2 (and Y is an arbitrary normed linear space), in particular if X = `p; 1 < p 2. This result is new also for real functions. We also prove that if X does not admit such equivalent norm, then there exists a Banach space Y (Y = `1(?) with card(?) = norm(X ), where norm(X ) denotes the minimal cardinality of a norming subset of SX ) and a C 1;1 mapping F : X ! Y (which is even a quadratic mapping) which is not d.c. Using this fact and a new observation that F ' is d.c. whenever F : X ! Y is a C 1;1 mapping and ' : (0; 1) ! X is d.c., we obtain a negative solution to Problem 6 of [31] (cf. the end of Section 2). In Section 3 we consider the following question: Let X; Y be Banach spaces, C be an open convex subset of X and let F : C ! Y be a d.c. mapping with a Lipschitz control function f . Is then F Lipschitz on C ? Note that, for d.c. functions (i.e. when Y = R ), the above question can be reformulated in the following way: Let f be a d.c. function on C , which admits

8


decompositions

f = c 1 ? l1 ; f = l 2 ? c 2 ;

where c1 ; c2 ; l1 ; l2 are continuous convex functions and l1 ; l2 are moreover Lipschitz. Is then f Lipschitz? (Indeed, if l is a Lipschitz control function for f , then we can put l1 = l2 = l. If f has the above decompositions, then l := l1 + l2 is a Lipschitz control function for f .) We show that the answer is positive if C is bounded or if C contains (a translate of) a nonempty open convex cone. A simple example shows that the answer is negative if X = R 2 , Y = R and C = (0; 1) R . We know no complete characterization of the sets C , for which the answer is positive. In Section 4 we present two simple examples. The rst one shows that the classes of (locally) order d.c. and weakly d.c. mappings are not stable with respect to compositions. In Section 5 we consider the situation when X; Y are Banach spaces, A X; B Y are open sets and F : A ! B is a locally d.c. bijection. We observe that, under some additional assumptions, X and Y are linearly isomorphic.

2. When all C 1;1 mappings are d.c.? If X and Y are normed linear spaces and A X is a (nonempty) open set, we denote by C 1;1(A; Y ) the set of all C 1;1 mappings from A into Y . Recall that F : A ! Y is a C 1;1 mapping if its Frechet derivative F 0(x) exists at each point x 2 A and the mapping F 0 is Lipschitz on A. The following proposition was proved in [31]. Proposition 5 ([31], Proposition 1.11). Let A be an open convex set in a Hilbert space and Y be a normed linear space. Then any mapping F 2 C 1;1(A; Y ) is d.c. with the control function Lip(F 0)k k2. It is natural to ask whether analogous statement holds if the Hilbert space is replaced by another normed linear space X . In this section we provide a characterization of such spaces X . Let us start recalling the de nition of the modulus of convexity of a normed linear space X : X (") = inf f1 ? kx+2 yk : kxk = kyk = 1 and kx ? yk "g; (0 < " 2): The next lemma follows from Bynum's results [7]. Though Bynum considered Banach spaces only, his proofs work for normed linear spaces as well. Lemma 6. For a normed linear space X , the following two conditions are equivalent: (a) X has modulus of convexity of power type 2, i.e. there exists a constant a > 0 such that X (") a"2 for each " 2 (0; 2];


9

(b) X satis es LWPL (the lower weak parallelogram law), i.e. there exists a constant b > 0 such that 2kxk2 + 2kyk2 kx + yk2 + bkx ? yk2 for all x; y 2 X . Let BX [BX0 ] denote the closed [open] unit ball of a normed linear space X .

Proposition 7. For a normed linear space (X; k k), the following three con-

ditions are equivalent: (i) X admits an equivalent norm whose modulus of convexity is of power type 2; (ii) X admits an equivalent norm satisfying LWPL; (iii) there exist r > 0, c > 0, and a? continuous (convex) function f : rBX0 ! R x+y 1 1 such that 2 f (x) + 2 f (y) ? f 2 ckx ? yk2 whenever x; y 2 rBX0 .

Proof. By Lemma 6, (i) and (ii) are equivalent. Let j j be an equivalent norm norm on X , satisfying LWPL with a constant b > 0 (as in Lemma 6). There exists > 0 such that j j k k. Direct calculation shows that (iii) holds with f (x) = jxj2, c = b4 , and an arbitrary r > 0. To complete the proof, it suces to show that (iii) implies (i). Let f be the function from (iii). Since (iii) remains valid after adding to f an arbitrary continuous convex function, we can (and do) suppose that f (0) = 0 and f 0 on rBX0 (by adding an appropriate continuous ane function) and, moreover, that f (x) = f (?x) for all x 2 rBX0 (by adding the continuous convex function g(x) = f (?x)). Since f is locally Lipschitz (cf. [18]), we can take a smaller r > 0 in such a way that f is Lipschitz on rBX0 . Denote L = Lip(f ). Fix an arbitrary 2 (0; 4cr2) and put

C = f x 2 rBX0 : f (x) g: Substituting y = ?x in the inequality of (iii), one obtains that f (x) 4ckxk2 ? for all x 2 rBX0 . It follows easily that C 4c 1=2 BX rBX0 . Consequently, C is closed in X . The set C , being a closed convex bounded symmetric set with nonempty interior in X , is the unit ball of an equivalent norm j j on X (j j is the Minkowski functional of C ). Let > 0 be such that, for all x 2 X ,

?1jxj kxk jxj: The properties of f easily imply that f (x) = whenever jxj = 1. (Indeed, for any x 6= 0, the function t 7! f (tx) is strictly increasing on [0; r=kxk).) We are going to show that (X; jj) has modulus of convexity of power type 2. Let jxj = jyj = 1 and z = x+2 y . Let z 2 X be such that z = jzjz and jz j = 1.

10

Then we have


ckx ? yk2 21 f (x) + 21 f (y) ? f (x + y)=2 ? = ? f (z) = f (z ) ? f jzjz ? L 1 ? jzj kz k L (1 ? jzj : ?

Consequently, 1 ? jzj L c kx ? yk2 (X;jj)(") a"2 for all " 2 (0; 2].

c L 3

jx ? yj2. Thus, for a = L c 3 , we get

By Lp() we mean a space Lp( ; ; ), where is a set, is a -algebra of subsets of , and : ! [0; 1] is a measure. As an example needed in the sequel, let us see which Lp() (1 < p < 1) admit renormings as above. (Obviously, in nite-dimensional spaces L1 () or L1() do not admit renormings with modulus of convexity of power type 2 since they are not re exive.)

Proposition 8. (a) (Hanner [12]) For 1 < p 2, the canonical norm on every nontrivial Lp () space has modulus of convexity of power type 2. (b) For 2 < p < 1, no in nite-dimensional Lp() space admits an equivalent norm whose modulus of convexity is of power type 2.

Proof of (b). Consider an in nite-dimensional Lp () with p > 2. If it had an equivalent norm with modulus of convexity of power type 2, then Lp() would have cotype 2 (cf. Theorem 1.e.16 in [16], or Theorem A.7 in [3]). But it is well-known that this is not the case (cf. [16], p. 73).

Lemma 9. Let X; Y be normed linear spaces, and A X be an open convex set. Then, for each F 2 C 1;1 (A; Y ) and all x; y 2 A,

1

F (x) +

2

1 F (y) ? F x + y 2 2

0

Lip(8F ) kx ? yk2:

Proof. First, recall that

(2.2) F (v) ? F (u) =

Z 1 0

?

F 0 u + t(v ? u) (v ? u) dt

This follows easily applyingthe formula '(1) ? '(0) = ? ping '(t) = F u + t(v ? u) .

whenever u; v 2 A. R1 0 0 ' (t) dt

to the map-


11

Consider x; y 2 A and F 2 C 1;1(A; Y ). For simplicity, let us denote w = y ? x, z = x+2 y and L = Lip(F 0). Then, using (2.2), we obtain

1

?

F (x) + 1 F (y ) ? F x+y = 1 F (y ) ? F (z ) ? F (z ) ? F (x) 2 2 2 2

R 1 0?

R 1 0 ? 1 = 2 0 F z + t(y ? z) (y ? z) dt ? 0 F x + t(z ? x) (z ? x) dt

R 1 0 0 F (z + tw=2) 1 R1 2 0 L z x w=2

= 21

k ? kk

? F 0(x + tw=2) (w=2) dt

k dt = L8 kwk2:

Recall that a mapping Q : X ! Y is a continuous quadratic mapping iff there exists a continuous symmetric bilinear mapping B : X X ! Y such that Q(x) = B (x; x) for all x 2 X . For such Q, we denote kQk = sup kQ(x)k: x2BX

Then kQk is nite and kQ(x)k kQkkxk2 for all x 2 X . Recall also that the norm kB k of a bilinear mapping B : P Q ! S (where P; Q; S are normed linear spaces) is de ned as kB k = supfkB (p; q)k : p 2 BP ; q 2 BQg: The norm kB k is nite i B is continuous and kB (p; q)k kB kkpkkqk for each p 2 P; q 2 Q. Observation 10. Let Q : X ! Y be a continuous quadratic mapping between normed linear spaces X and Y . Then Q 2 C 1;1(X ; Y ) and

1

1 x + y

= 1 kQ(x ? y )k kQk kx ? y k2 : (2.3)

Q(x) + Q(y) ? Q 2 2 2 4 4 Proof. Let B be a continuous symmetric bilinear mapping for which Q(x) = B (x; x). The equality in (2.3) follows by direct calculation. Another direct calculation gives, for any x; v 2 X , kQ(x + v) ? Q(x) ? 2B (x; v)k = kQ(v)k kQkkvk2: Consequently, Q is Frechet dierentiable at each x 2 X , and Q0(x) = 2B (x; ). Obviously, Q0 is linear. It is easy to see that kQ0k 2kB k. By linearity, Q0 is (2kB k)-Lipschitz. Let us denote SX := f x 2 X : kx k = 1 g. A set ? SX is said to be norming if there exists p > 0 such that supf jz (x)j : z 2 ? g pkxk for all x 2 X (equivalently, if this \sup" de nes an equivalent norm on X ). Now we are ready to state the main result of the present section.

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Theorem 11. For a normed linear space X , the following conditions are

equivalent. (i) X admits an equivalent norm whose modulus of convexity is of power type 2. (ii) There exists a continuous convex function f : X ! R such that, for every open convex set A X and every normed linear space Y , each mapping F 2 C 1;1(A; Y ) is d.c. with the control function Lip(F 0) f . (ii') There exists a continuous convex function f : X ! R such that, for every open convex set A X , each function F 2 C 1;1 (A; R ) is d.c. with the control function Lip(F 0) f . (iii) There exists a continuous convex function f : X ! R such that, for every normed linear space Y , each continuous quadratic mapping Q : X ! Y is d.c. with the control function kQk f . (iii') There exists a continuous convex function f : X ! R such that each continuous quadratic form Q : X ! R is d.c. with the control function kQk f . (iv) For every open convex set A X and every normed linear space Y , each mapping F 2 C 1;1 (A; Y ) is d.c. (v) For every normed linear space Y , each continuous quadratic mapping Q : X ! Y is d.c. Proof. Suppose (i) is satis ed. Then, by Proposition 7, there exists an equivalent norm j j on X such that, for some b > 0, bjx ? yj2 2jxj2 + 2jyj2 ? jx + yj2: Let > 0 be such that kxk jxj for every x 2 X . If F is as in (ii), we have by Lemma 9

1 ? Lip(F 0 )

F (x) + 1 F (y ) F x+y x y 2 2 Lip(8 F 0) x y 2 2 2 2 8 ? ?

2 Lip(F 0 ) 2+2y 2 2 ) = Lip(F 0 ) 1 f (x) + 1 f (y ) f x+y ; (2 x x + y 8b 2 2 2 2 where f (x) = 2b x 2 . We have proved that (i) implies (ii). In the same way,

?

j j j j ?j jj

j

k ? k

j ? j ?

using 2.3 instead of2 Lemma 9, one obtains the implication (i) ) (iii) (this time with f (x) = b jxj2). The implications (ii) ) (ii0), (ii) ) (iv), (iii) ) (iii0) and (iii) ) (v) are obvious. Moreover, (iv) ) (v) follows from Observation 10. It remains to show that each of the three conditions (ii'), (iii'), (v) implies (i). By Proposition 7, it suces to show that each of these three conditions implies existence of r; c > 0 and a continuous convex function f : rBX0 ! R such that 1 f (x) + 1 f (y) ? f x + y ckx ? yk2: 2 2 2


13

Let (ii') or (iii') be satis ed. Put A = BX0 . Given x; y 2 A, nd x 2 X such that kx k = 1 and jx(x ? y)j = kx ? yk. Consider the continuous bilinear form Bx;y (u; v) = x (u)x(v) and the corresponding continuous quadratic 2 form Qx;y (u) = Bx;y (u; u) = x (u) . It is easy to see that, for any x; y 2 A, kQx;y k1 = 1 and Qx;y 2 C 1;1(A; R ) with Lip(Q0x;y ) = 2 on A (cf. Observation 10). Hence, by our assumption, there exists a common control function f : A ! R for all quadratic forms Qx;y (x; y 2 A). By Observation 10 and the de nition of Q, we have 1 f (x) + 1 f (y ) ? f ? x+y 1 Q (x) + 1 Q (y ) ? Q ? x+y x;y 2 2 2 2 x;y 2 x;y 2 1 1 2 = 4 jQx;y (x ? y)j = 4 kx ? yk whenever x; y 2 A, what we needed. Now, let (v) be satis ed. Let ? SX be any norming set (e.g., ? = SX ). De ne a mapping Q : X ! `1(?) by

Q(x) = z (x) 2

z 2?

:

Then Q is a continuous quadratic mapping (comming from the continuous symmetric? bilinearmapping B (x; y) = (z (x) z (y))z2? ). By Observation 10, Q 2 C 1;1 A; `1(?) . By (v), Q is controlled by a continuous convex function f on A. Hence, for any x; y 2 A, we get ? x+y 1 ? x+y 1 1 1

2 f (x) + 2 f (y ) ? f 2 2 Q(x) + 2 Q(y ) ? Q 2 = 41 kQ(x ? y)k = 41 supf jz(x ? y)j2 : z 2 ? g

p42 kx ? yk2:

It remains open whether the conditions (i)-(v) in Theorem 11 are equivalent to the following condition: (vi) For every open convex set A X , each function F 2 C 1;1(A; R ) is d.c. It is still possible that (vi) holds for every normed linear space X . Let us denote by norm(X ) the minimal cardinality of a norming subset of SX . (It is easy to see that norm(X ) dens(X ) dens(X ) holds for every X .) Analyzing the proof of Theorem 11, we can state the following strengthening of the implication (v) ) (i). Lemma 12. Let X be a normed linear space such that no equivalent norm on X has modulus of convexity of power ? type 2. Then there exists a continuous quadratic mapping Q : X ! `1 (?) card(?) = norm(X ) such that Q is d.c. on no open convex subset of X .

14


Proof. Let ? SX be a norming set with card(?) = norm(X ). It follows easily from the proof of (v) ) (i) in Theorem 11, that the continuous quadratic mapping ? Q : X ! `1(?); Q(x) = [z (x)]2 z2? is d.c. on no neighborhood of 0. Let B : X 2 ! `1(?) be the continuous symmetric bilinear mapping that generates Q. For any xed a 2 X , the identity Q(a + x) = Q(a) + 2B (x; a) + Q(x) (x 2 X ) easily implies that Q is d.c. on no neighborhood of a. The above lemma and the following Proposition 14 will give a solution to Problem 6 in [31]. Proposition 14 is an easy consequence of the following simple lemma which is an obvious generalization of the classical result on the product of two real functions. Lemma 13. Let P; Q; S be normed linear spaces and let B : P Q ! S be a continuous bilinear mapping. Let a; b 2 R and u : [a; b] ! P , v : [a; b] ! Q have nite variation. Then also w : t 7! B (u(t); v(t)) has nite variation on [a; b]. Moreover ? Vab w kB k kuk1Vabv + kvk1Vabu : Proof. Consider a partition a = t0 < t1 < < tn = b of the interval [a; b]. Our statement immediately follows from the following inequalities n?1 X i=1

kw(ti+1)?w(ti)k = kB kkuk1

n?1 X i=1

n?1 X i=1

kB (u(ti+1); v(ti+1)?v(ti))+B (u(ti+1)?u(ti ); v(ti))k

kv(ti+1) ? v(ti)k + kB kkvk1

n?1 X i=1

ku(ti+1) ? u(ti)k:

Proposition 14. Let X; Y be normed linear spaces, A X be an open convex set and let F 2 C 1;1 (A; Y ). Let I R be an open interval and let ' : I ! A be a d.c. mapping. Then F ' is a d.c. mapping. Proof. Consider an arbitrary interval [a; b] I . By Theorem 3, it is sub 0 0 cient ? 0 to prove 0 Va (F ')+ < 1: The chain rule easily gives (F ')+ (t) = F ('(t)) ('+(t)) for each t 2 [a; b]. Putting P := L(X; Y ), Q := X , S := Y and B (p; q) := p(q) for p 2 P; q 2 Q, we have that B : P Q ! S is bilinear and continuous (with kB k = 1). Put u(t) := F 0('(t)); v(t) := '0+(t): Since both F 0 and ' (cf. Theorem 4,a) are Lipschitz on [a; b], we obtain that u is Lipschitz on [a; b] and therefore Vabu < 1: By Theorem 3 we have Vabv < 1. To complete the proof, it is sucient to use Lemma 13.


15

Remark 1. It is easy to see that, under the assumptions of Proposition 14, Lemma 13 gives Vab (F ')0+ kF 0 'k1 Vab'0+(t) + Lip(')Lip(F 0)Vab ':

In [31], Problem 6, the following question was asked (cf. Introduction): Let X; Y be Banach spaces, A X be an open convex set and F : A ! Y be a mapping. If F ' is d.c. on (0; 1) whenever ' : (0; 1) ! A is d.c., is then F locally d.c. on A ? Our results imply an answer in negative to the above problem. Indeed, since `3 has no equivalent norm with modulus of convexity of power type 2 (Proposition 8), by Lemma 12 there exists a continuous quadratic mapping Q : `3 ! `1 which is locally d.c. at no point of `3. On the other hand, by Observation 10 and Proposition 14, Q ' is d.c. whenever ' : (0; 1) ! `3 is d.c. 3. Is a d.c. mapping Lipschitz if it has a Lipschitz control function?

This question arose quite naturally in [8] in the connection with discussions of dierent topologies on sets of d.c. mappings. Here we present generalizations of results of [8] (obtained by a dierent method). We will need the following simple and well-known facts on continuous convex functions on a Banach space X . Before their formulation, recall that (for a real function g on X ) the symbol 0 g+(x; v) denotes the one-sided directional derivative of g at x in the direction v and, for a continuous convex function f de ned on an open convex subset of X , @f (x) X is the subdierential (the set of all subgradients) of f at a point x. The open ball with center x and radius r will be denoted by B (x; r). Recall also that a mapping F (between metric spaces) is K -Lipschitz if Lip(F ) K , where Lip(F ) is the (least) Lipschitz constant of F . The statements of the following lemma are simple and well-known. Lemma 15. Let f be a continuous convex function de ned on an open convex subset C of a Banach space X . Then the following statements hold. (i) If x; v 2 X , then f+0 (x; v) = maxfx (v) : x 2 @f (x)g and therefore ?f+0 (x; ?v) x (v) f+0 (x; v) for each x 2 @f (x). (ii) If kx k K whenever x 2 C and x 2 @f (x), then f is K -Lipschitz on C. (iii) If x 2 C; > 0; v 2 X and y := x + v 2 C , then ?f+0 (x; ?v) f+0 (x; v) ?f+0 (y; ?v) f+0 (y; v): (iv) If B (x0 ; r) C and jf (x)j K for x 2 B (x0 ; r)

16


then f is (4K=r)-Lipschitz on B (x0 ; r=2): For (i) see e.g. [18]; (ii) follows easily from the de nition of the subdierential; (iii) is just a reformulation of well-known facts about one-sided derivatives of convex functions of one real variable, applied to g(t) = f (x + tv); (iv) follows e.g. from the proof of Theorem 41.B in [20]. We will need also the following easy lemma on real d.c. functions. Lemma 16. Let C be an open subset of a Banach space X , x0 ; x 2 C and let g be a d.c. function on C with a control function f . Further suppose that f is K -Lipschitz on C and g is K0 -Lipschitz on a neigbourhood of x0 . Then (i) jg+0 (x; x ? x0 )j (K0 +2K )kx ? x0 k, jg+0 (x; x0 ? x)j (K0 +2K )kx ? x0 k and (ii) jg(x)j jg(x0)j + (K0 + 2K )kx ? x0 k: Proof. Note that (ii) is a simple consequence of (i) (e.g. by the classical Dini theorem) and (i) follows easily from the proof of Theorem 2.3 of [31]. However, for the sake of completness, we give an alternative simple proof. We can and do suppose f (x0 ) = 0. Clearly jf+0 (x; x ? x0 )j K kx ? x0k; jf+0 (x; x0 ? x)j K kx ? x0 k; jf+0 (x0 ; x ? x0 )j K kx ? x0k; jf+0 (x0 ; x0 ? x)j K kx ? x0 k; jg+0 (x0; x ? x0 )j K kx ? x0 k and jf (x)j K kx ? x0 k: Since h := g + f is convex, we obtain (using Lemma 15,(iii)) ?(K + K0)kx ? x0 k h0+(x0 ; x ? x0 ) ?h0+ (x; x0 ? x) = ?g+0 (x; x0 ? x) ? f+0 (x; x0 ? x) h0+(x; x ? x0 ) = g+0 (x; x ? x0 )+ f+0 (x; x ? x0 ) and consequently (3.4) g+0 (x; x ? x0 ) ?(2K + K0)kx ? x0 k;

(3.5) ?g+0 (x; x0 ? x) ?(2K + K0)kx ? x0 k: Using Lemma 15,(i) we obtain (3.6) g(x) + f (x) = h(x) h(x0 ) + h0+(x0 ; x ? x0 ) g(x0 ) ? (K + K0)kx ? x0 k and (3.7) ?g(x) jg(x0)j + (2K + K0)kx ? x0 k: Applying the inequalities (3.4), (3.5), (3.7) to g~ := ?g (we can do it, since g~ + f is also convex on C and g~ is K0-Lipschitz on a neighbourhood of x0 ), the inequalities (i),(ii) easily follow. Our main lemma is the following one.


17

Lemma 17. Let C be an open convex subset of a Banach space X and let g be a d.c. function on C with a K -Lipschitz control function f . Let B (a; r) C and jg(t)j M for t 2 B (a; r). Then, for each R > 0, g is Lipschitz on C \ B (a; R) with the constant (9K + 4M=r)(8R=r + 1): Proof. We can and do suppose f (a) = 0. We know that h := g +f is continuous and convex on C . We are going to estimate the Lipschitz constant of h on C \ B (a; R). Fix arbitrary x 2 C \ B (a; R) and x 2 @h(x). To estimate kx k, consider an arbitrary v 2 X; kvk = 1 and put x0 = a + rv=4. For t 2 B (a; r), we have clearly jh(t)j M + jf (t)j M + Kr. Lemma 15,(iv) now implies that h is Lipschitz on B (a; r=2) with the constant 4(M + Kr)=r = 4M=r +4K . Therefore g is Lipschitz on B (a; r=2) with the constant 4M=r + 5K . By Lemma 16,(i) we obtain the inequalities jg+0 (x; x ? x0 )j (7K + 4M=r) kx ? x0 k; jg+0 (x; x0 ? x)j (7K + 4M=r) kx ? x0 k; jh0+(x; x ? x0)j (8K + 4M=r) kx ? x0 k; jh0+(x; x0 ? x)j (8K + 4M=r) kx ? x0 k:

Therefore (using Lemma 15,(i)), we have jx (x ? x0 )j (8K + 4M=r) kx ? x0 k: Quite similarly (applying Lemma 16 to x~0 := a) we obtain jx(x ? a)j (8K + 4M=r) kx ? ak: Consequently we obtain j(rv=4; x)j = jx(x0 ? a)j (8K + 4M=r) (kx ? x0k + kx ? ak) (8K + 4M=r) (2R + r=4) and therefore jx (v)j (8K + 4M=r) (8R=r + 1) =: L: Thus kx k L and we obtain by Lemma 15,(ii) that h is L-Lipschitz on C \ B (a; R). Since f is K -Lipschitz, the assertion of the lemma follows. Theorem 18. Let X; Y be Banach spaces, C be an open convex subset of X and let F : C ! Y be a d.c. mapping with a Lipschitz control function f . Suppose that one of the following two conditions is satis ed: (i) C is bounded. (ii) C contains (a translate of) a nonempty open convex cone. Then the mapping F is Lipschitz on C .

18


Proof. Consider arbitrary points u; v 2 C and put D := (F (u) ? F (v)) =ku ? vk: Find y 2 Y ; kyk = 1 such that y(D) = kDk. Clearly f controls the d.c. function g := y F and (3.8) kDk = y(D) = g(u) ? g(v) :

ku ? vk

First suppose that the condition (i) is satis ed. Choose a 2 C and (using continuity of F ) numbers r; M; R > 0 such that kF (t)k M for t 2 B (a; r) and C B (a; R). Then clearly also kg(t)k M for t 2 B (a; r) and thus Lemma 17 implies that g is Lipschitz on C with the constant (9K + 4M=r)(8R=r + 1): By (3.8) we obtain that F is Lipschitz on C with the same constant. Now suppose that (ii) holds. Thus we can nd x0 2 C , a direction v; kvk = 1, and > 0 such that, for each t > 0, B (x0 + tv; t) C . Since F is locally Lipschitz (see Theorem 4) we choose K0 > 0 such that F (and therefore also g) is K0 -Lipschitz on a neigbhourhood of x0 . Let K be a Lipschitz constant of f . Now choose an arbitrary x 2 C and nd t > 0 so large that, for a := x0 +tv, the inequality kx ? ak < 2ka ? x0 k holds. Putting r := t, we have ka ? x0 k = r= and R := 2r= > kx ? ak: Without any loss of generality we can suppose that F (x0) = 0 (and therefore g(x0) = 0). Applying Lemma 16,(ii) (with x = z), we obtain that, for each z 2 B (a; r), jg(z)j (K + 2K )kz ? x k (K + 2K )(r + r ) = (K + 2K )r 1 + :

0 Now Lemma 17 gives that g is Lipschitz on C \ B (a; R) with the constant 4( K 16 0 + 2K )(1 + ) L := 9K + +1 : Since x was an arbitrary element of C and x 2 C \ B (a; R), we obtain that g is locally, and therefore also globally, L-Lipschitz on C . By (3.8) we obtain that F is Lipschitz on C with the same constant. 0

0

0

Note that Dostal [8] proved the part (i) of the previous theorem for Hilbert spaces and (ii) for C = X . We are not able to characterize the sets C for which Theorem 18 holds. The following simple example shows that it does not hold for every C . Proposition 19. There exists a d.c. non-Lipschitz function g on the strip C := (0; 1) R , which has a Lipchitz control function f .


19

Proof. We put g(x; y) := x log(1 + jyj) and f (x) = x2 ? 2 log(1 + jyj) + 3jyj for (x; y) 2 C . It is immediate to check that f is Lipschitz but g is not Lipschitz on C . It remains to show that f controls g; i.e. that both g + f and ?g + f are convex on C . The functions f g are C 2 on C+ := (0; 1) (0; 1). Their second order partial derivatives and their Hessian D on C+ are: (f g)xx(x; y) = 2; (f g)xy (x; y) = (1 + y)?1;

(f g)yy (x; y) = (2 x)(1 + y)?2; D = 2(2 x)(1 + y)?2 ? (1 + y)?2 = (3 2x)(1 + y)?2 > 0: Thus the functions f g are convex on C+, since their Hessian matrices are positively de nite at all points of C+. By symmetry, the functions f g are convex also on C? := (0; 1) (?1; 0): To prove that these functions are convex on the whole C , it is sucient to prove that (3.9) (f g)0+((a; 0); v) + (f g)0+((a; 0); ?v) 0 for each a 2 (0; 1) and each v 2 R 2 . Indeed, then we can clearly conclude that f g is convex on each segment S C and therefore it is convex on C . Since a direct easy computation gives that, if v = (v1 ; v2) and v2 0, (f g)0+((a; 0); v) = 2av1 +(1 a)v2 ; (f g)0+((a; 0); ?v) = ?2av1 +(1 a)v2 ; the condition (3.9) holds and we are done. 4. Two examples concerning order d.c. mappings As already mentioned in Introduction, a generalization of Hartman's composition theorem does not hold for weakly d.c. mappings and order d.c. mappings. We present here a simple counterexample (in [31] it is noted only that validity of such generalizations is doubtfull). In the following we consider the usual coordinate-wise ordering of `2. Proposition 20. There exists a mapping f : R ! `2, which is both order d.c. and weakly d.c., such that g := kk`2 f is not d.c. (and thus, since g is a real function, g is neither weakly d.c. nor order d.c. ). Proof. Take any sequence fan gn2N such that an & 0 and for n 2 N de ne (

for t 2 R n (an+1; an); fn(t) = 0an ?an+1 for t = an +2an+1 ; 2n

20


and anely on the closures of the two remaining (open) intervals. Further de ne 8 > for x an+1; n (x ? an+1) for x 2 (an+1; an); :2 an ?an+1 for x an; n (x ? an ) + n and ( an +an+1 0 for x ; 2 ? kn(x) = 2 an +an+1 an +an+1 for x 2 : n x? 2 It is easy to see that fn = hn ? kn and all hn; kn are convex. Now put f (x) = (f1(x); f2 (x); : : : ); h(x) = (h1 (x); h2(x); : : : ); k(x) = (k1 (x); k2(x); : : : ): It is easy to see to see that f; g; h are mappings R ! `2 and f = h ? k. Since hn ; kn are clearly (2=n)-Lipschitz, we easily obtain that h; k are Lipschitz convex operators and f is order d.c. It is easy to verify that f is also weakly d.c. Indeed, if c 2 (`2 ), we have c = (c; :), where c = fcngn2N 2 `2 and (:; :) is the scalar product on `2. Then X X X X + ? + ? c f (x) = cn hn (x) + cn kn(x) ? cn kn(x) + cn hn(x) is d.c. on R . P Clearly g(x) = 1 n=1 fn (x) for x 2 R (the series has, at any point, at most 0 0 an +an+1 ) = ?1=n. one nonzero term!) and therefore W" 0 g+ (an+1 ) = 1=n and g+ ( 2 Consequently, the variation ?" g+ = 1 for each " > 0, and so, by Theorem 3,(ii), we get that g is d.c. on no neighbourhood of 0. The following example is an improvement of the Example 6.1 from [31] which shows that there exists a mapping f : R ! `2 which is order d.c., weakly d.c., but is not d.c. Proposition 21. Let X be a separable Banach space. Then there exists a mapping F : X ! `2 which is order d.c., weakly d.c., and which is d.c. on no nonempty convex open set C X . Proof. Let f : R ! `2 , h : R ! `2 , k : R ! `2 be mappings from the preceeding example. Thus h; k are Lipschitz convex operators, f = h ? k is order d.c., weakly d.c. and kf k is d.c. on no neighbourhood of 0. The composition theorem Theorem 2 implies that also f is d.c. on no neighbourhood of 0. Let fxngn2N be a dense subset of XP . Fix a 2 X and a 2 X such that kak = ka k = a (a) = 1. Denote Y = n `2 `2 . Note that Y is canonically isometrically and order isomorphic to `2 . For n 2 N , let n : Y ! `2 be the projection of Y on the n-th coordinate and let En be the embedding En : `2 ! Y such that n En = id and k En = 0 for k 6= n.


21

For n 2 N consider the mappings Fn; Hn; Kn : X ! Y de ned for x 2 X by the formulas Fn = En(f (a(x ? xn ))); Kn = En(k(a(x ? xn ))); Hn = En (h(a(x ? xn ))): Since f; h; k are Lipschitz, we can choose constants cn > 0 so that cn max fkFn(x)k; kHn(x)k; kKn(x)kg n?2 for each x 2 B (0; n): P Finally, de ne the P mapping F : X ! Y asPF (x) = n cnFn(x) for x 2 X , and similarly H (x) = n cnHn(x); K (x) = n cnKn(x). Since the series de ning F; H; K converge uniformly on each bounded subset of X , the mappings F; H; K are well de ned and continuous. Since H and K are order convex as sums of such mappings, we obtain that F = H ? K is order d.c. Proceeding as in the proof that f is weakly d.c. (c.f. proof of Proposition 20), it is easy to verify that F is also weakly d.c. Now consider a nonempty open convex set C X and suppose that F is d.c. on C . Find n 2 N such that xn 2 C , put A(t) := ta + xn and choose > 0 such that A((?; )) C . By Lemma 1.5 from [31] (or Theorem 2 and Theorem 3,(ii)) we obtain that, for each n, n F A is d.c. on (?; ). But n F A(t) = cnf (t) for t 2 (?; ), which is a contradiction with the properties of f . 5. Which Banach spaces are \d.c. isomorphic"? Let X; Y be Banach spaces, which are Lipschitz isomorphic; i.e. there exists a Lipschitz bijection F : X ! Y such that F ?1 is also Lipschitz. The natural question, whether then X and Y are linearly isomorphic, led to an interesting and deep theory (cf. [3]). (Note only that the answer is negative in general, but it is still possible that it is positive for separable spaces X; Y .) We will observe that two well-known simple facts of this theory easily imply that the answer is positive under the additional assuptions that F (and F ?1) is (are) also d.c. and under some assumptions about X and Y . More precisely, we prove the following result. Theorem 22. Suppose that X; Y are Banach spaces, A X; B Y are nonempty open convex sets and F : A ! B is a bijective locally d.c. mapping such that F ?1 is locally Lipschitz on B and moreover (i) X is an Asplund space or (ii) X; Y are weak Asplund spaces and F ?1 is locally d.c. Then X and Y are linearly isomorphic. Proof. We will need the following well-known easy facts (cf. [3], p. 170). Let X; Y be Banach spaces, A X; B Y open sets, F : A ! B be a bi-Lipschitz bijection and a 2 A. If

22


() F has a Frechet derivative at a, or ( ) F has a Gâteaux derivative at a and F ?1 has a Gâteaux derivative at F (a), then X and Y are linearly isomorphic. By Theorem 4,a we can suppose that F is Lipschitz on A and F ?1 is Lipschitz on B . Suppose rst that (i) holds. Then Theorem 4,c gives a 2 A for which () holds and thus we are done. Now suppose (ii) holds. Let N1 A and N2 B be the sets of Gâteaux non-dierentiability points of F and F ?1, respectively. By Theorem 4,c and Banach category theorem (saying that a set is of the rst category if it is locally of the rst category), we obtain that N1; N2 , and consequently also F ?1(N2 ), are rst category sets. Now it is sucient to choose a 2 A n (N1 [ F ?1(N2 )) and to apply ( ).

Acknowledgment. The research of the second author was partially sup-

ported by the Ministero dell'Universita e della Ricerca Scienti ca e Tecnologica of Italy. The research of the third author was partially supported by the grant GAC R 201/00/0767 from the Grant Agency of Czech Republic and partially supported by the grant MSM 113200007 from the Czech Ministry of Education. References [1] A. D. Alexandrov, On surfaces represented as the dierence of convex functions, Izv. Akad. Nauk. Kaz. SSR 60, Ser. Math. Mekh. 3 (1949), 3{20 (in Russian). [2] , Surfaces represented by the dierences of convex functions, Dokl. Akad. Nauk SSSR (N.S.) 72 (1950), 613{616 (in Russian). [3] Y. Benyamini, J. Lindenstrauss, Geometric Nonlinear Functional Analysis, vol. 1, Colloquium Publications (American Mathematical Society), Providence, Rhode Island, 2000. [4] M. Cepedello Boiso, Approximation of Lipschitz functions by -convex functions in Banach spaces, Israel J. Math. 106 (1998), 269{284. [5] , On regularization in superre exive Banach spaces by in mal convolution formulas, Studia Math. 129 (1998), 265{284. [6] , Two characterizations of super-re exive Banach spaces by the behaviour of dierences of convex functions, preprint, 1999. [7] W. L. Bynum, Weak parallelogram laws for Banach spaces, Canad. Math. Bull. 19 (1976), 269{275. [8] M. Dostal, Spaces of delta-convex mappings, Thesis, Charles University Prague, 1993. [9] J. Duda, On inverses of -convex mappings, Preprint Series, Charles University, Prague, KMA{2000{30. [10] R. Ger, Stability aspects of delta{convexity, In: Stability mappings of Hyers{Ulam type (Th. M. Rassias and J. Tabor eds.), Hadronic Press, Florida, 1994, pp. 99{109.

ON D.C. FUNCTIONS AND MAPPINGS [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34]

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Department of Mathematics, University of Missouri{Columbia, Columbia, MO 65211, USA Dipartimento di Matematica, Universita degli Studi, Via C. Saldini 50, 20133 Milano, Italy Department of Mathematical Analysis, Charles University, Sokolovska 83, 186 00 Prague 8, Czech Republic

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