Increasing functions and inverse Santaló inequality for unconditional functions

Abstract

Let $\phi: {\mathbb{R}}^n\to {\mathbb{R}}\cup\{+\infty\}$ be a convex function and $\mathcal{L}\phi$ be its Legendre tranform. It is proved that if $\phi$ is invariant by changes of signs, then $\int e^{-\phi}\int e^{-\mathcal{L}\phi} \ge 4^n$. This is a functional version of the inverse Santaló inequality for unconditional convex bodies due to J. Saint Raymond. The proof involves a general result on increasing functions on $\mathbb{R}^{n} \times \mathbb{R}^n$ together with a functional form of Lozanovskii’s lemma. In the last section, we prove that for some c > 0, one has always $\int e^{-\phi}\int e^{-\mathcal{L}\phi} \ge c^n$. This generalizes a result of B. Klartag and V. Milman.