The hamburger theorem

We generalize the ham sandwich theorem to d +1 measures on R-d as follows. Let mu(1), mu(2),..., mu(d+1) be absolutely continuous finite Borel measures on R-d. Let omega(i) = mu(i) (R-d) for i is an element of [d + 1], omega = min{omega(i) : i is an element of[d + 1]) and assume that Sigma(d+1)(j=1) omega(j)= 1. Assume that omega(i) = 1/(d + 1). As a consequence we obtain that every (d + 1)-colored set of nd points in R-d such that no color is used for more than n points can be partitioned into n disjoint rainbow (d - 1)-dimensional simplices. (c) 2017 Elsevier B.V. All rights reserved.