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

Abstract : 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.
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https://hal-upec-upem.archives-ouvertes.fr/hal-03115432
Contributor : Matthieu Fradelizi <>
Submitted on : Tuesday, January 19, 2021 - 4:20:26 PM
Last modification on : Friday, February 5, 2021 - 3:32:04 AM
Long-term archiving on: : Tuesday, April 20, 2021 - 7:49:07 PM

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### Identifiers

• HAL Id : hal-03115432, version 1
• ARXIV : 2101.08065

### Citation

Matthieu Fradelizi, Elie Nakhle. The functional form of Mahler conjecture for even log-concave functions in dimension $2$. 2021. ⟨hal-03115432⟩

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