Sections of the difference body

M. Rudelson

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arXiv · math/9812008

Sections of the difference body

M. Rudelson

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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.

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M. Rudelson. 1998-12-01. Sections of the difference body. https://arxiv.org/abs/math/9812008

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