The colourful simplicial depth conjecture

Pauline Sarrabezolles (1)

1 CERMICS - Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique

Given $d+1$ sets of points, or colours, $\S_1,\ldots,\S_{d+1}$ in $\R^d$, a {\em colourful simplex} is a set $T\subseteq\bigcup_{i=1}^{d+1}\S_i$ such that $|T\cap \S_i|\leq 1$, for $i=1,\ldots,d+1$. The colourful \cara{} theorem states that, if $\zero$ is in the convex hull of each $\S_i$, then there exists a colourful simplex $T$ containing $\zero$ in its convex hull. In 2006, Deza, Huang, Stephen, and Terlaky ({\em Colourful simplicial depth}, Discrete Comput. Geom., {\bf 35}, 597--604 (2006)) conjectured that, actually, when $|\S_i|=d+1$ for all $i=1,\ldots,d+1$, there are always at least $d^2+1$ colourful simplices containing $\zero$ in their convex hulls. We prove this conjecture with the help of combinatorial objects called octahedral systems.

Mots clés: colourful Carathéodory theorem; colourful simplicial depth; octahedral systems

Domaines: Mathématique discrète [cs.DM]

Pauline Sarrabezolles. The colourful simplicial depth conjecture. 2014. ⟨hal-00943550v2⟩