Quotients of Continuous Convex Functions on Nonreflexive Banach Spaces
Volume 55 / 2007
Bulletin Polish Acad. Sci. Math. 55 (2007), 211-217
MSC: 46B10, 46B03.
DOI: 10.4064/ba55-3-3
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 also gives a stronger version of Klee's result concerning renormings of nonreflexive spaces and non-norm-attaining functionals.
