  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 $i\in
[d+1]$, $\omega=\min\{\omega_i; i\in [d+1]\}$ and assume that $\sum_{j=1}^{d+1}
\omega_j=1$. Assume that $\omega_i \le 1/d$ for every $i\in[d+1]$. Then there
exists a hyperplane $h$ such that each open halfspace $H$ defined by $h$
satisfies $\mu_i(H) \le (\sum_{j=1}^{d+1} \mu_j(H))/d$ for every $i \in [d+1]$
and $\sum_{j=1}^{d+1} \mu_j(H) \ge \min(1/2, 1-d\omega) \ge 1/(d+1)$. 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.
