Colorful simplicial depth, Minkowski sums, and generalized Gale transforms
Combinatorics
2016-07-04 v1 Metric Geometry
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.
Cite
@article{arxiv.1607.00347,
title = {Colorful simplicial depth, Minkowski sums, and generalized Gale transforms},
author = {Karim Adiprasito and Philip Brinkmann and Arnau Padrol and Pavel Paták and Zuzana Patáková and Raman Sanyal},
journal= {arXiv preprint arXiv:1607.00347},
year = {2016}
}
Comments
17 pages, 3 figures