Colorful simplicial depth, Minkowski sums, and generalized Gale transforms

TL;DR

This paper establishes tight upper bounds on colorful simplicial depth and related Minkowski sum properties using combinatorial topology, confirming several longstanding conjectures in discrete geometry.

Contribution

It proves a tight upper bound on colorful simplicial depth and introduces colorful Gale transforms linking configurations to Minkowski sums, resolving multiple conjectures.

Findings

Proved a tight upper bound on colorful simplicial depth.

Introduced colorful Gale transforms connecting configurations and Minkowski sums.

Resolved conjectures by Deza et al. (2006) and Burton (2003).

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. (2006). 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 (2003) in the theory of normal surfaces.