Quadratically Many Colorful Simplices

The colorful Carathéodory theorem asserts that if X1, X2, . . . , Xd+1 are sets in R, each containing the origin 0 in its convex hull, then exists a set S ⊆ X1 ∪ · · · ∪ Xd+1 with |S ∩ Xi| = 1 for all i = 1, 2, . . . , d + 1 and 0 ∈ conv(S) (we call conv(S) a colorful covering simplex). Deza, Huang, Stephen and Terlaky proved that if the Xi are in general position with respect to 0 (consequently, each Xi has at least d + 1 points), then there are at least 2d colorful covering simplices, and they constructed an example with no more than d + 1 such simplices. Under the same assumption, we show that there are at least 1 5 d(d + 1) colorful covering simplices, thus determining the order of magnitude. We also obtain a lower bound of 3d, which is better for small d, and in particular, together with a parity argument it settles the case d = 3, where the minimum possible number of colorful covering simplices is 10.