The Functional Form of Mahler’s Conjecture for Even Log-Concave Functions in Dimension 2

Abstract Let $\varphi :{\mathbb {R}}^{n}\rightarrow {\mathbb {R}}\cup \{+\infty \}$ be an even convex function and ${\mathcal {L}}{\varphi }$ be its Legendre transform. We prove the functional form of Mahler’s conjecture concerning the functional volume product $P(\varphi )=\int e^{-\varphi }\int e^{-{\mathcal {L}}\varphi }$ in dimension 2: we give the sharp lower bound of this quantity and characterize the equality case. The proof uses the computation of the derivative in $t$ of $P(t\varphi )$ and ideas due to Meyer [16] for unconditional convex bodies, adapted to the functional case by Fradelizi and Meyer [6] and extended for symmetric convex bodies in dimension 3 by Iriyeh and Shibata [11] (see also [4]).