Inverse Additive Problems for Minkowski Sumsets II

Abstract

The Brunn-Minkowski Theorem asserts that ${\mu }_d(A+B{)}^{1/d}\ge {\mu }_d(A{)}^{1/d}+{\mu }_d(B{)}^{1/d}$ for convex bodies $A,B\subseteq {\mathbb{R}}^d$, where ${\mu }_d$ denotes the $d$-dimensional Lebesgue measure. It is well-known that equality holds if and only if $A$ and $B$ are homothetic, but few characterizations of equality in other related bounds are known. Let $H$ be a hyperplane. Bonnesen later strengthened this bound by showing $${\mu }_d(A+B)\ge (M^{1/(d-1)}+N^{1/(d-1)}{)}^{d-1}(\frac{{\mu }_d(A)}{M}+\frac{{\mu }_d(B)}{N}),$$ where $M=\sup \{{\mu }_{d-1}((\mathbf{x}+H)\cap A)\mid \mathbf{x}\in {\mathbb{R}}^d\}$ and $N=\sup \{{\mu }_{d-1}((\mathbf{y}+H)\cap B)\mid \mathbf{y}\in {\mathbb{R}}^d\}$. Standard compression arguments show that the above bound also holds when $M={\mu }_{d-1}(\pi (A))$ and $N={\mu }_{d-1}(\pi (B))$, where $\pi$ denotes a projection of ${\mathbb{R}}^d$ onto $H$, which gives an alternative generalization of the Brunn-Minkowski bound. In this paper, we characterize the cases of equality in this later bound, showing that equality holds if and only if $A$ and $B$ are obtained from a pair of homothetic convex bodies by `stretching' along the direction of the projection, which is made formal in the paper. When $d=2$, we characterize the case of equality in the former bound as well.

References in corpus (1)

• Inverse Additive Problems for Minkowski Sumsets I

Cited by in corpus (2)

• Inverse Additive Problems for Minkowski Sumsets I
• Additive combinatorics with a view towards computer science and cryptography: An exposition