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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-ba55-3-3,
author = {P. Holick\'y and O. F. K. Kalenda and L. Vesel\'y and L. Zaj\'\i \v cek},
title = {Quotients of Continuous Convex Functions on Nonreflexive Banach Spaces},
journal = {Bulletin of the Polish Academy of Sciences. Mathematics},
volume = {55},
year = {2007},
pages = {211-217},
zbl = {1135.46005},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-ba55-3-3}
}
P. Holický; O. F. K. Kalenda; L. Veselý; L. Zajíček. Quotients of Continuous Convex Functions on Nonreflexive Banach Spaces. Bulletin of the Polish Academy of Sciences. Mathematics, Tome 55 (2007) pp. 211-217. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-ba55-3-3/