The hamburger theorem

We generalize the ham sandwich theorem to $d+1$ measures in ${\mathbb{R}}^d$ as follows. Let ${\mu }_1,{\mu }_2,\dots ,{\mu }_{d+1}$ be absolutely continuous finite Borel measures on ${\mathbb{R}}^d$. Let ${\omega }_i={\mu }_i({\mathbb{R}}^d)$ for $i\in [d+1]$,$\omega=\min{\omega_i; i\in [d+1]}$andassumethat$\sum_{j=1}^{d+1} \omega_j=1$\mathrm{.}Assumethat$\omega_i \le 1/d$forevery$i\in[d+1]$. Then there exists a hyperplane $h$ such that each open halfspace $H$ defined by $h$ satisfies ${\mu }_i(H)\le ({\sum }_{j=1}^{d+1}{\mu }_j(H))\mathrm{/}d$ for every $i\in [d+1]$ and ${\sum }_{j=1}^{d+1}{\mu }_j(H)\ge \min (1\mathrm{/}2,1-d\omega )\ge 1\mathrm{/}(d+1)$. 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.