The hamburger theorem

We generalize the ham sandwich theorem to d+1 measures on Rd as follows. Let μ1,μ2,…,μd+1 be absolutely continuous finite Borel measures on Rd. Let ωi=μi(Rd) for i∈[d+1], ω=min⁡{ωi;i∈[d+1]} and assume that ∑j=1d+1ωj=1. Assume that ωi≤1/d for every i∈[d+1]. Then there exists a hyperplane h such that each open halfspace H defined by h satisfies μi(H)≤(∑j=1d+1μj(H))/d for every i∈[d+1] and ∑j=1d+1μj(H)≥min⁡{1/2,1−dω}≥1/(d+1). As a consequence we obtain that every (d+1)-colored set of nd points in Rd such that no color is used for more than n points can be partitioned into n disjoint rainbow (d−1)-dimensional simplices.