Quotients of continuous convex functions on nonreflexive Banach spaces

TL;DR

This paper demonstrates that on nonreflexive Banach spaces, there exist positive continuous convex functions whose reciprocals are not d.c., revealing a characterization of reflexivity via quotients of convex functions.

Contribution

It constructs specific convex functions on nonreflexive spaces showing the non-d.c. nature of their reciprocals, and characterizes reflexivity through quotients of convex functions.

Findings

Existence of convex functions with non-d.c. reciprocals on nonreflexive spaces

Reflexivity characterized by quotients of convex functions being d.c.

Strengthened results related to Klee's renorming theorem.

Abstract

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 gives also a stronger version of Klee's result concerning renormings of nonreflexive spaces and non-norm-attaining functionals.