A Further Generalization of the Colourful Carathéodory Theorem

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 Si 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