Estimates for measures of lower dimensional sections of convex bodies

2015/12/28 by Chasapis, Giorgos, Giannopoulos, Apostolos, Liakopoulos, Dimitris-Marios · 1 citation
#FOS: Mathematics #Functional Analysis (math.FA) #Metric Geometry (math.MG)

paper · doi:10.48550/arxiv.1512.08393

Abstract

We present an alternative approach to some results of Koldobsky on measures of sections of symmetric convex bodies, which allows us to extend them to the not necessarily symmetric setting. We prove that if K is a convex body in \mathbb Rⁿ with 0∈ \rm int(K) and if μ is a measure on \mathbb Rⁿ with a locally integrable non-negative density g on \mathbb Rⁿ, then μ(K)≤ (c√(n-k) )^kmax_F∈ G_n,n-kμ(K∩ F)⋅ |K|^(k)/(n) for every 1≤ k≤ n-1. Also, if μ is even and log-concave, and if K is a symmetric convex body in \mathbb Rⁿ and D is a compact subset of \mathbb Rⁿ such that μ(K∩ F)≤ μ(D∩ F) for all F∈ G_n,n-k, then μ(K)≤ (ckL_n-k )^kμ(D), where L_s is the maximal isotropic constant of a convex body in \mathbb R^s. Our method employs a generalized Blaschke-Petkantschin formula and estimates for the dual affine quermassintegrals.

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