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

2021/01/20 by Matthieu Fradelizi, Fradelizi, Matthieu, Elie Nakhle +1
Biochemistry, Genetics and Molecular Biology · Mathematics · Medicine · #FOS: Mathematics #Functional Analysis (math.FA) #Geometric Analysis and Curvature Flows #Markov Chains and Monte Carlo Methods #Metric Geometry (math.MG) #Pharmacological Effects of Medicinal Plants #Point processes and geometric inequalities #Prion Diseases and Protein Misfolding

paper · pdf · doi:10.48550/arxiv.2101.08065

openalex publication_date 2021/01/20 · openalex created_date 2022/09/14 · openalex updated_date 2026/07/28

Abstract

Let Φ : R n → R ∪ +∞ be an even convex function and LΦ be its Legendre transform. We prove the functional form of Mahler conjecture concerning the functional volume product P (Φ) = e --Φ e --LΦ 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 (tΦ) 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]).

Citations

• The deficit in the Gaussian log-Sobolev inequality and inverse Santalo inequalities

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