Sections of the difference body

Abstract

Let $K$ be an $n$-dimensional convex body. Define the difference body by $$K-K=\{x-y\mid x,y\in K\}.$$ 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 $x\in R^n$, such that $$vol((K-K)\cap F)\le C^m(\min (n/m,\sqrt{m}){)}^m\cdot vol(K\cap (F+x)),$$ for some absolute constant $C$. We show that for small dimensions of $F$ this estimate is exact up to a multiplicative constant.

10 pages, AMSTeX