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.

11 pages, 2 figures; a new proof of Theorem 8, extended concluding remarks