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Delta-convex mappings between Banach spaces and applications

L. L. Veselý;  L. Zajíček — 1989

We investigate delta-convex mappings between normed linear spaces. They provide a generalization of functions which are representable as a difference of two convex functions (labelled as 5-convex or d.c. functions) and are considered in many articles. We show that delta-convex mappings have many good differentiability properties of convex functions and the class of them is very stable. For example, the class of locally delta-convex mappings is closed under superpositions and (in some situations)...

Quotients of Continuous Convex Functions on Nonreflexive Banach Spaces

P. Holický;  O. F. K. Kalenda;  L. Veselý;  L. Zajíček — 2007

Bulletin of the Polish Academy of Sciences. Mathematics

On each nonreflexive Banach space X there exists a positive continuous convex function f such that 1/f is not a d.c. function (i.e., a difference of two continuous convex functions). This result together with known ones implies that X is reflexive if and only if each everywhere defined quotient of two continuous convex functions is a d.c. function. Our construction also gives a stronger version of Klee's result concerning renormings of nonreflexive spaces and non-norm-attaining functionals.

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