The union of colorful simplices spanned by a colored point set

A simplex spanned by a colored point set in Euclidean d-space is colorful if all vertices have distinct colors. The union of all full-dimensional colorful simplices spanned by a colored point set is called the colorful union  . We show that for every d∈Nd∈N, the maximum combinatorial complexity of the colorful union of n   colored points in RdRd is between Ω(n(d−1)2)Ω(n(d−1)2) and O(n(d−1)2logn). For d=2d=2, the upper bound is known to be O(n)O(n), and for d=3d=3 we present an upper bound of O(n4α(n))O(n4α(n)), where α(⋅)α(⋅) is the extremely slowly growing inverse Ackermann function. We also prove several structural properties of the colorful union. In particular, we show that the boundary of the colorful union is covered by O(nd−1)O(nd−1) hyperplanes, and the colorful union is the union of d+1d+1 star-shaped polyhedra. These properties lead to efficient data structures for point inclusion queries in the colorful union.