Complete Kneser Transversals
Résumé
Let $k,d,\lambda\geqslant1$ be integers with $d\geqslant\lambda $. In 2010, the following function was introduced:
$m(k,d,\lambda)\overset{\mathrm{def}}{=}$ the maximum positive integer $n$ such that every set of $n$ points (not necessarily in general position) in $\mathbb{R}^{d}$ has the property that the convex hulls of all $k$-sets have a common transversal $(d-\lambda)$-plane.
It turns out that $m(k, d,\lambda)$ is strongly connected with other interesting problems, for instance, the chromatic number of Kneser hypergraphs and a discrete version of Rado's centerpoint theorem.
In the same spirit, we introduce a natural discrete version $m^*$ of $m$ by considering the existence of complete Kneser transversals. We study the relation between them and give a number of lower and upper bounds of $m^*$ as well as the exact value in some cases. The main ingredient for the proofs are Radon's partition theorem as well as oriented matroids tools. By studying the so-called alternating oriented matroid we provide the asymptotic behavior of the function $m^*$ for the family of cyclic polytopes.
Origine : Fichiers produits par l'(les) auteur(s)