The hamburger theorem

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