The functional form of Mahler conjecture for even log-concave functions in dimension $2$

Matthieu Fradelizi (1, 2) , Elie Nakhle (3, 2)

1 Université Gustave Eiffel
2 LAMA - Laboratoire Analyse et de Mathématiques Appliquées
3 UPEC UP12 - Université Paris-Est Créteil Val-de-Marne - Paris 12

Let $\varphi: \mathbb{R}^n\to \mathbb{R}\cup\{+\infty\}$ be an even convex function and $\mathcal{L}{\varphi}$ be its Legendre transform. We prove the functional form of Mahler 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 for unconditional convex bodies, adapted to the functional case by Fradelizi-Meyer and extended for symmetric convex bodies in dimension 3 by Iriyeh-Shibata.

Mots clés: Mahler conjecture; Blaschke-Santalo inequality; functional form; Legendre transform; equipartitions; log-concave functions

Domaines: Analyse fonctionnelle [math.FA]; Géométrie métrique [math.MG]

Matthieu Fradelizi, Elie Nakhle. The functional form of Mahler conjecture for even log-concave functions in dimension $2$. International Mathematics Research Notices, 2023, 2023 (12), pp.10067-10097. ⟨10.1093/imrn/rnac120⟩. ⟨hal-03115432⟩