Colorful Simplicial Depth, Minkowski Sums, and Generalized Gale Transforms

Karim A. Adiprasito, Philip Brinkmann, Arnau Padrol*, Pavel Paták, Zuzana Patáková, Raman Sanyal

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

3 Scopus citations

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.

Original languageAmerican English
Article number184
Pages (from-to)1894-1919
Number of pages26
JournalInternational Mathematics Research Notices
Volume2019
Issue number6
DOIs
StatePublished - 22 Mar 2019

Bibliographical note

Publisher Copyright:
© The Author(s) 2017.

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