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

DOI10.1093/IMRN/RNAC120arXiv2101.08065WikidataQ113819037 ScholiaQ113819037MaRDI QIDQ6161402

Unnamed Author, Matthieu Fradelizi

Publication date: 27 June 2023

Published in: IMRN. International Mathematics Research Notices (Search for Journal in Brave)

Abstract: Let $Phi$ : R n $ightarrow$ 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]).

Full work available at URL: https://arxiv.org/abs/2101.08065

Mathematics Subject Classification ID

Convexity of real functions of several variables, generalizations (26B25)

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