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

TL;DR

This paper establishes the sharp lower bound for the functional volume product of even convex functions in two dimensions, extending Mahler's conjecture to the functional setting and characterizing the equality case.

Contribution

It proves the functional form of Mahler's conjecture for even log-concave functions in dimension 2, providing the sharp bound and equality characterization.

Findings

Sharp lower bound for the functional volume product in dimension 2

Characterization of the equality case for the bound

Extension of Mahler's conjecture to the functional setting

Abstract

Let $\Phi$ : R n $\rightarrow$ R $\cup$ {+$\infty$} be an even convex function and L$\Phi$ be its Legendre transform. We prove the functional form of Mahler conjecture concerning the functional volume product P ($\Phi$) = e --$\Phi$ e --L$\Phi$ 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$\Phi$) and ideas due to Meyer [M] for unconditional convex bodies, adapted to the functional case by Fradelizi-Meyer [FM2] and extended for symmetric convex bodies in dimension 3 by Iriyeh-Shibata [IS] (see also [FHMRZ]).