A Further Generalization of the Colourful Carathéodory Theorem

Abstract

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 in the union of the S_i such that the intersection of S with any S_i is of cardinality at most 1. The convex hull of a colourful set S is called a colourful simplex. Barany's colourful Carath\'eodory theorem asserts that if the origin 0 is contained in the convex hull of each of the S_i, 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 union S_j for i