The hamburger theorem

Abstract

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.Comment: 10 pages, 2 figures; corrected a serious error in the proof of Theorem 8, further minor change

Near equipartitions of colored point sets

“…We show that then there exist n pairwise disjoint convex sets, each of them containing k of the points, and each of them containing points of both colors.We also show that if P is a set of n(d + 1) points in general position in R d colored by d colors with at least n points of each color, then there exist n pairwise disjoint d-dimensional simplices with vertices in P , each of them containing a point of every color.These results can be viewed as a step towards a common generalization of several previously known geometric partitioning results regarding colored point sets.In this note, we prove two results concerning partitions of colored point sets. We conjecture a common generalization of these results, as well as various other related results and conjectures [1,2,10]. First we establish some basic terminology.Definitions.…”

“…In this note, we prove two results concerning partitions of colored point sets. We conjecture a common generalization of these results, as well as various other related results and conjectures [1,2,10]. First we establish some basic terminology.…”

“…Another recent example in this direction is the following generalization of Theorem 5, due to Kano and Kynčl [10].…”

“…Theorem 2 confirms Conjecture 3 for k − 1 = m = d ≥ 2. The proof of Theorem 2 is based on the continuous ham sandwich theorem and a special discretization argument [5,10] and it is given in Section 4.…”

“…Theorem 7 (Kano-Kynčl, [10]). Let d ≥ 2 and n ≥ 1 be integers, and let X be a (d + 1)colored point set in general position in R d .…”