On the global shape of continuous convex functions on Banach spaces

Abstract

We make some remarks on the global shape of continuous convex functions defined on a Banach space $Z$. Among other results we prove that if $Z$ is separable then for every continuous convex function $f:Z\to\mathbb{R}$ there exist a unique closed linear subspace $Y_f$ of $Z$ such that, for the quotient space $X_f :=Z/Y_{f}$ and the natural projection $\pi:Z\to X_f$, the function $f$ can be written in the form $$ f(z)=\varphi(\pi(z)) +\ell(z) \textrm{ for all } z\in Z, $$ where $\ell_{f}\in X^{*}$ and $\varphi:X_f\to\mathbb{R}$ is a convex function such that $\lim_{t\to\infty}\varphi(x+tv)=\infty$ for every $x, v\in X_f$ with $v\neq 0$. This kind of result is generally false if $Z$ is nonseparable (even in the Hilbertian case $Z=\ell_{2}(\Gamma)$ with $\Gamma$ an uncountable set).