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.