A note on the reduction of the slicing problem to centrally symmetric convex bodies

2022/10/17 by Javier Martín-Goñi, Martín-Goñi, Javier · 1 citation
Mathematics · Medicine · #Drug Transport and Resistance Mechanisms #FOS: Mathematics #Functional Analysis (math.FA) #Metric Geometry (math.MG) #Pharmacological Effects of Medicinal Plants #Point processes and geometric inequalities

paper · pdf · doi:10.48550/arxiv.2210.09137

openalex publication_date 2022/10/17 · openalex created_date 2022/10/20 · openalex updated_date 2026/07/28

Abstract

In this paper, we obtain the best possible value of the absolute constant C such that for every isotropic convex body K ⊆ ℝⁿ the following inequality (which was proved by Klartag and reduces the hyperplane conjecture to centrally symmetric convex bodies) is satisfied: L_K≤ CL_K_n+2(g_K). Here L_K denotes the isotropic constant of K, g_K its covariogram function, which is log-concave, and, for any log-concave function g, K_n+2(g) is a convex body associated to the log-concave function g, which belongs to a uniparametric family introduced by Ball. In order to obtain this inequality, sharp inclusion results between the convex bodies in this family are obtained whenever g satisfies a better type of concavity than the log-concavity, as g_K is, indeed (1)/(n)-concave.

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