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Quotients of continuous convex functions on nonreflexive Banach spaces

Holicky P., Kalenda O. F., Vesely L., Zajícek L.

Bulletin of the Polish Academy of Sciences. Mathematics

2007

Vol. 55, no 3

211-217

On each nonreflexive Banach space X there exists a positive continuous convex function ƒ such that 1/ƒ 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.