Sections of the Difference Body

Abstract. 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 ∈ \bf R n , such that $$ \mbox{\rm vol} ((K-K) \cap F) \le C^m \left ( \min \left ( \frac{n}{m}, \sqrt{m} \right ) \right )^m \cdot \mbox{\rm vol} (K \cap (F+x)),$$ for some absolute constant C . We show that for small dimensions of F this estimate is exact up to a multiplicative constant.