On the differentiation of convex functions in finite and infinite dimensional spaces

Abstract

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On sets of non-differentiability of Lipschitz and convex functions

“…By induction, the proposition follows. Proposition 3.6 and the result of [19] mentioned in Introduction immediately imply the main result of this section (in which, of course, we can write N F (f ) instead of N G(f )).…”

“…Another definition we need is that of the notion (see [19], [20]) of d.c. (that is, delta-convex) surfaces of finite codimension. First recall the definition (see [18]) of d.c. mappings.…”

“…It is not probable that the generalization of Corollary 3.7 holds in an infinitedimensional space. On the other hand, using [19,Theorem 2] and the fact that every d.c. function is locally Lipschitz, it is easy to see that N G(f ) is σ-lower porous whenever f is a continuous convex function on a separable Banach space. However, it seems to be probable that the following problem has a negative answer.…”

“…Note that a full characterization of these sets is given in [10] for X = ¡ n , but even the case of a Hilbert space X is open. In Section 3 we present some consequences of the above mentioned result of [19] and of P. Hartman's theorem on superposition of deltaconvex mappings. In particular, we show that the set N G(f ) is σ-strongly lower porous for each convex function on ¡ n .…”

“…If f is a continuous convex function on a separable Banach space X, then the set N G(f ) of Gâteaux non-differentiability points of f can be covered by countably many of d.c. (that is, delta-convex) hypersurfaces (see [19,Theorem 2] or [3,Theorem 4.20]); moreover, this result is the optimal one, if we are interested in the smallness of sets N G(f ) (for a continuous convex f ) only. Note that a full characterization of these sets is given in [10] for X = ¡ n , but even the case of a Hilbert space X is open.…”