Abstrakt
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 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 = min{1/2, 1 - d*omega} >= 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.