A further generalization of the colourful Carath\'eodory theorem

TL;DR

This paper extends the colourful Carathéodory theorem by providing a broader sufficient condition for the existence of colourful simplices containing the origin, along with algorithms and proofs, including a graph-based approach in the plane.

Contribution

It generalizes the conditions for colourful simplices containing 0 and introduces an algorithm and graph-based proof methods.

Findings

Generalized the sufficient condition for colourful simplices containing 0.

Provided an algorithm to find such simplices under the new condition.

Showed that any condition implying the existence of a colourful simplex also guarantees multiple such simplices.

Abstract

Given $d+1$ sets, or colours, $S_1, S_2,...,S_{d+1}$ of points in $\mathbb{R}^d$, a {\em colourful} set is a set $S\subseteq\bigcup_i S_i$ such that $|S\cap S_i|\leq 1$ for $i=1,...,d+1$. The convex hull of a colourful set $S$ is called a {\em colourful simplex}. B\'ar\'any's colourful Carath\'eodory 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 $S_i\cup S_j$ for $1\leq i< j \leq d+1$ by Arocha et al. and by Holmsen et al. We further generalize the sufficient condition and obtain new colourful Carath\'eodory 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.