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Minkowski和集II的逆加法问题

Inverse Additive Problems for Minkowski Sumsets II

G. A. Freiman, D. J. Grynkiewicz, O. Serra, Y. Stanchescu

发表信息：
Journal of Geometric Analysis, January 2013, Vol. 23, Issue 1, pp 395-414
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摘要：

Brunn-Minkowski定理断言，对于凸体$A,\,B\subseteq \R^d$，有$μ_d(A+B)^{1/d}\geq μ_d(A)^{1/d}+μ_d(B)^{1/d}$，其中$μ_d$表示$d$维勒贝格测度。众所周知，等式成立当且仅当$A$和$B$是位似的，但在其他相关的界限中，等式的特征很少。设$H$是一个超平面。Bonnesen后来通过证明$$μ_d(A+B)\geq (M^{1/(d-1)}+N^{1/(d-1)})^{d-1}(\frac{μ_d(A)}{M}+\frac{μ_d(B)}{N}),$$（其中$M=\sup\{μ_{d-1}((\mathbf x+H)\cap A)\mid \mathbf x\in \R^d\}$和$N=\sup\{μ_{d-1}((\mathbf y+H)\cap B)\mid \mathbf y\in \R^d\}$）加强了这种结合。标准压缩参数表明，当$M=μ_{d-1}(π(A))$和$N=μ_{d-1}(π(B))$，其中$π$表示$\mathbb R^d$到$H$的投影时，上述界限也成立，这给出了Brunn-Minkowski界限的另一种推广。在本文中，我们的特点是平等的情况下，在这后界，表明平等持有的当且仅当$A$和$B$是从一对位似凸体的“拉伸”沿投影的方向，这是正式的文件。当$d=2$，我们的特点是平等的情况下，在前界以及。

英文摘要：

The Brunn-Minkowski Theorem asserts that $μ_d(A+B)^{1/d}\geq μ_d(A)^{1/d}+μ_d(B)^{1/d}$ for convex bodies $A,\,B\subseteq \R^d$, where $μ_d$ 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 $$μ_d(A+B)\geq (M^{1/(d-1)}+N^{1/(d-1)})^{d-1}(\frac{μ_d(A)}{M}+\frac{μ_d(B)}{N}),$$ where $M=\sup\{μ_{d-1}((\mathbf x+H)\cap A)\mid \mathbf x\in \R^d\}$ and $N=\sup\{μ_{d-1}((\mathbf y+H)\cap B)\mid \mathbf y\in \R^d\}$. Standard compression arguments show that the above bound also holds when $M=μ_{d-1}(π(A))$ and $N=μ_{d-1}(π(B))$, where $π$ 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.

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