Remarks on an inequality of Rogers and Shephard

Abstract

A classical inequality of Rogers and Shephard states that if K is a centered convex body of volume 1 in Rn then 1 6 g(K, k;F ) := ( volk(PF (K)) voln−k(K ∩ F⊥) )1/k 6 (n k )1/k 6 cn k for every F ∈ Gn,k, where c > 0 is an absolute constant. We show that if K is origin symmetric and isotropic then, for every 1 6 k 6 n − 1, a random F ∈ Gn,k satisfies c1L −1 K √ n/k 6 g(K, k;F ) 6 c2 √ n/k (logn)LK with probability greater than 1 − e−k, where LK is the isotropic constant of K and c1, c2 > 0 are absolute constants.