The colourful simplicial depth conjecture

Given d + 1 sets of points, or colours, S 1 , ? , S d + 1 in R d , a colourful simplex is a set T ? ? i = 1 d + 1 S i such that | T ? S i | ? 1 for all i ? { 1 , ? , d + 1 } . The colourful Caratheodory 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 et al. (2006) 3 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.

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