Convex functions and the Radon-Nikodým property
The authors give geometrical properties of lower semi-continuous convex functions and obtain a characterization of Banach spaces with the Radon-Nikodým property in terms of regularity of convex functions defined on the space. Their main result is the following: A Banach space \(X\) has the Radon-Nikodým property if and only if every proper convex function \(f:X\to {]}-\infty,+\infty]\) which is lower semi-continuous and lower bounded shares one of the following properties: (1) There is a denting point for \(f\). (2) There is a strongly exposed point in the set \(S(f,\alpha)=\{(x,r)| \, f(x)\leq r\leq\alpha\}\), for some real \(\alpha\). (3) There is a (strong) minimum for \(f-x'\), for some \(x'\) in the dual of \(X\). (4) There is a weak rotundity point for \(f\). (5) There exists \(\alpha\) such that there is a denting point in the set \(S(f,\alpha)\). (6) There is a weak minimum for \(f\). See \textit{W. Schachermayer}, \textit{A. Sersouri} and \textit{E. Werner} [Isr. J. Math. 65, 225--257 (1989; Zbl 0686.46011)] and \textit{R. Deville}, \textit{G. Godefroy} and \textit{V. Zizler} [``Smoothness and renormings in Banach spaces'' (Pitman Mon. Surv. Pure Appl. Math. 64, Longman, Harlow) (1993; Zbl 0782.46019)] for related results.
scientific article; zbMATH DE number 2031053