Estimates for measures of lower dimensional sections of convex bodies

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 R-n with 0 is an element of int(K) and mu is a measure on R-n with a locally integrable non-negative density g on R-n, then mu(K) <= (c root n-k)(k) max(F is an element of Gn,n-k) mu(K boolean AND F) . vertical bar K vertical bar(k/n) for every 1 <= k <= n-1. Also, if mu is even and log-concave, and if K is a symmetric convex body in R-n and D is a compact subset of R-n such that mu(K boolean AND F) < mu(D boolean AND F) for all F is an element of G(n,n-k;) then mu(K) <= (ckL(n-k))(k) mu(D), where L-s is the maximal isotropic constant of a convex body in R-s. Our method employs a generalized Blaschke-Petkantschin formula and estimates for the dual affine quermassintegrals. (C) 2016 Elsevier Inc. All rights reserved.