Session B2 - Graph Theory and Combinatorics
July 14, 16:00 ~ 16:25
Colorful simplicial depth, Minkowski sums, and generalized Gale transforms
Université Pierre et Marie Curie, France - firstname.lastname@example.org.
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.
Joint work with Karim Adiprasito (Hebrew University of Jerusalem), Philip Brinkmann (Freie Universität Berlin), Pavel Paták (Hebrew University of Jerusalem), Zuzana Patáková (Hebrew University of Jerusalem), Raman Sanyal (Goethe Universität Frankfurt).