Characterization of reflexivity by convex functions (Q739900)

Let \(X\) be a real Banach space and \(D\) a nonempty convex subset of \(X\). A function \(f:D\to\mathbb{R}\) is called 2R (w2R) if, for every bounded sequence \((x_n)\) in \(D\), the condition \(\lim_{m,n\to\infty}[(f(x_n)+f(x_m))/2-f((x_n+x_m)/2)]=0\) implies the norm convergence of \((x_n)\) (respectively, the weak convergence). The Banach space \(X\) is called 2R (w2R) if \(f(x)=\|x\|^2\) is 2R (w2R). It was shown by \textit{E. Odell} and \textit{Th. Schlumprecht} [J. Am. Math. Soc. 11, No. 1, 175--188 (1998; Zbl 0888.46006)] that a separable Banach space is reflexive iff it admits a 2R renorming. In general, a Banach space is reflexive iff it admits a w2R renorming; cf. [\textit{P. Hájek} and \textit{M. Johanis}, J. Funct. Anal. 211, No. 1, 163--172 (2004; Zbl 1055.46005)]. The main result of the paper asserts that a Banach space \(X\) is reflexive iff there exist an open convex set \(D\subset X\) and a continuous w2R convex function \(f:D\to \mathbb{R}\). If \(X\) is separable, then the reflexivity is equivalent to the existence of a continuous 2R convex function defined on some open convex subset of \(X\). The proof is based on the fact (Lemma 4) that the existence of a continuous w2R function on a closed convex set \(K\subset X\) implies the weak compactness of \(K\). It was shown in [\textit{L.-X. Cheng} et al., Acta Math. Sin., New Ser. 14, No. 1, 47--56 (1998; Zbl 1006.46050)] that a Banach space \(X\) is superreflexive iff there exists a continuous uniformly convex function \(f\) defined on some open convex subset of \(X\).