Sections of the Difference Body 3

Let K be an n-dimensional convex body. Define the difference body by K − K = {x − y | x, y ∈ K}. We estimate the volume of the section of K − K by a linear subspace F via the maximal volume of sections of K parallel to F. We prove that for any m-dimensional subspace F there exists x ∈ R n , such that vol ((K − K) ∩ F) ≤ C m min n m , √ m m · vol (K ∩ (F + x)) , for some absolute constant C. We show that for small dimensions of F this estimate is exact up to a multiplicative constant.