Colorful Simplicial Depth, Minkowski Sums, and Generalized Gale Transforms

TY - JOUR

T1 - Colorful Simplicial Depth, Minkowski Sums, and Generalized Gale Transforms

AU - Adiprasito, Karim A.

AU - Brinkmann, Philip

AU - Padrol, Arnau

AU - Patak, Pavel

AU - Patakova, Zuzana

AU - Sanyal, Raman

PY - 2019/3/22

Y1 - 2019/3/22

N2 - 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.

AB - 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.

U2 - 10.1093/imrn/rnx184

DO - 10.1093/imrn/rnx184

M3 - Journal article

SN - 1073-7928

VL - 2019

SP - 1894

EP - 1919

JO - International Mathematics Research Notices

JF - International Mathematics Research Notices

IS - 6

ER -