Title: The hamburger theorem

Abstract:
We generalize the ham sandwich theorem to $d+1$ measures on $\mathbb{R}^d$ 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 $i\in[d+1]$ (that is, the measures are \emph{balanced} in
$\mathbb{R}^d$).
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.

Further straightforward generalization of the ham sandwich theorem to $d+2$
measures in $\mathbb{R}^d$ is not possible for $d \ge 4$.
We conjecture that for $d+i$ measures balanced in $\mathbb{R}^d$, where $1\le
i \le d-1$, a nontrivial balanced partition into at most $i+1$ convex parts
exists. A discrete version of this conjecture would imply a conjecture by Kano
and Suzuki that every $(d+i)$-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.

The talk is based on a paper coauthored by Mikio Kano.