A Further Generalization of the Colourful Carathéodory Theorem

Abstract

Given d+1 sets, or colours, $\mathbf{S}_{1},\mathbf{S}_{2},\ldots,\mathbf{S}_{d+1}$ of points in ${\mathbb{R}}^{d}$, a colourful set is a set $S \subseteq \bigcup _{i}\mathbf{S}_{i}$ such that $\vert S \cap \mathbf{S}_{i}\vert \leq 1$ for $i = 1,\ldots,d + 1$. 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 $i = 1,\ldots,d + 1$, 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 $\mathbf{S}_{i} \cup \mathbf{S}_{j}$ for 1≤i<j≤d+1 by Arocha etal. and by Holmsen etal. We further generalize the sufficient condition and obtain new colourful Carathéodory theorems. 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 any condition implying the existence of a colourful simplex containing 0 actually implies the existence of min _i |S _i |such simplices.