Colorful Simplicial Depth, Minkowski Sums, and Generalized Gale Transforms

Abstract

Abstract The colorful simplicial depth of a collection of $d+1$ finite sets of points in Euclidean $d$-space is the number of choices of a point from each set such that the origin is contained in their convex hull. We use methods from combinatorial topology to prove a tight upper bound on the colorful simplicial depth. This implies a conjecture of Deza et al. [7]. Furthermore, we introduce colorful Gale transforms as a bridge between colorful configurations and Minkowski sums. Our colorful upper bound then yields a tight upper bound on the number of totally mixed facets of certain Minkowski sums of simplices. This resolves a conjecture of Burton [6] in the theory of normal surfaces.