The colourful simplicial depth conjecture

Abstract

Given d + 1 sets of points, or colours, S 1 , . . . , S d+1 in R d , a colourful simplex is a set T ⊆ d+1 i=1 S i such that |T ∩ S i | ≤ 1, for all i ∈ {1, . . . , d + 1}. The colourful Carathéodory theorem states that, if 0 is in the convex hull of each S i , then there exists a colourful simplex T containing 0 in its convex hull. Deza, Huang, Stephen, and Terlaky (Colourful simplicial depth, Discrete Comput. Geom., 35, 597-604 (2006)) conjectured that, when |S i | = d + 1 for all i ∈ {1, . . . , d + 1}, there are always at least d 2 + 1 colourful simplices containing 0 in their convex hulls. We prove this conjecture via a combinatorial approach.