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

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...

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Bibliographic Details
Published inInternational mathematics research notices Vol. 2023; no. 12; pp. 10067 - 10097
Main Authors Fradelizi, Matthieu, Nakhle, Elie
Format Journal Article
LanguageEnglish
Published Oxford University Press (OUP) 12.06.2023
Subjects
Functional Analysis
Mathematics
Metric Geometry
Blaschke-Santalo inequality
functional form
log-concave functions
equipartitions
Mahler conjecture
Legendre transform
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Summary: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.
ISSN:1073-7928
1687-0247
DOI:10.1093/imrn/rnac120