汉堡定理
The hamburger theorem
摘要：
我们将火腿三明治定理推广到$\mathbb{R}^d$中的$d+1$度量，如下所示。设$μ_1,μ_2, \dots, μ_{d+1}$是$\mathbb{R}^d$上绝对连续的有限Borel测度.设$ω_i=μ_i(\mathbb{R}^d)$为$i\in [d+1]$，$ω=\min\{ω_i; i\in [d+1]\}$，并假设$\sum_{j=1}^{d+1} ω_j=1$。假设对于每个$i\in[d+1]$，都有$ω_i \le 1/d$。则存在一个超平面$h$，使得由$h$定义的每个开半空间$H$对每个$i \in [d+1]$和$\sum_{j=1}^{d+1} μ_j(H) \ge \min(1/2, 1-dω) \ge 1/(d+1)$都满足$μ_i(H) \le (\sum_{j=1}^{d+1} μ_j(H))/d$。因此，我们得到了$\mathbb{R}^d$中的$nd$点的每个$(d+1)$着色集，使得没有颜色用于超过$n$点可以被划分成$n$不相交的彩虹$(d-1)$维单形.
英文摘要：
We generalize the ham sandwich theorem to $d+1$ measures in $\mathbb{R}^d$ as follows. Let $μ_1,μ_2, \dots, μ_{d+1}$ be absolutely continuous finite Borel measures on $\mathbb{R}^d$. Let $ω_i=μ_i(\mathbb{R}^d)$ for $i\in [d+1]$, $ω=\min\{ω_i; i\in [d+1]\}$ and assume that $\sum_{j=1}^{d+1} ω_j=1$. Assume that $ω_i \le 1/d$ for every $i\in[d+1]$. Then there exists a hyperplane $h$ such that each open halfspace $H$ defined by $h$ satisfies $μ_i(H) \le (\sum_{j=1}^{d+1} μ_j(H))/d$ for every $i \in [d+1]$ and $\sum_{j=1}^{d+1} μ_j(H) \ge \min(1/2, 1-dω) \ge 1/(d+1)$. As a consequence we obtain that every $(d+1)$-colored set of $nd$ points in $\mathbb{R}^d$ such that no color is used for more than $n$ points can be partitioned into $n$ disjoint rainbow $(d-1)$-dimensional simplices.
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