Given d+1 sets, or colours, S_1,S_2,...,S_(d+1) of points in R^d, a colourful set is a set S containing at most one point from each S_i for i = 1,...,d+1. The convex hull of a colourful set S is called a colourful simplex. Bárány’s colourful Carathéodory theorem asserts that if the origin 0 is contained in the convex hull of S_i for i = 1,...,d +1, then there exists a colourful simplex containing 0. The sufficient condition for the existence of a colourful simplex containing 0 was generalized to 0 being contained in the convex hull of (S_i U S_j) for 1 ? i < j ? d+1 by Arocha et al. and by Holmsen et al. We further generalize the theorem by showing that a colourful simplex containing 0 exists under weaker conditions. A slightly stronger version of this new colourful Carathéodory theorem is also given. This result provides a short and geometric proof of the previous generalization of the colourful Carathéodory theorem. We also give an algorithm to find a colourful simplex containing 0 under the generalized condition. In the plane an alternative and more general proof using graphs is given. In addition, we observe that, in general, the existence of one colourful simplex containing 0 implies that the number of such simplices is at least the size of the smallest S_i. In other words, any condition implying the existence of a colourful simplex containing 0 actually implies that the number of such simplices is at least the size of the smallest S_i.

Joint work with Frédéric Meunier