Blocking codimension-one simplices on the moment curve

Abstract

We study $b_d(n)$, the minimum number of points needed to meet the relative interior of every $(d-1)$-simplex spanned by an $n$-point set in general position in ${\mathbb{R}}^d$. In the plane, this is the parameter from the Blocking Conjecture. We improve the best known general planar lower bound to $b_2(n)\ge \frac{41}{13}n-O\left(\frac{n}{\log n}\right)$. For $n$ points on the moment curve in even dimension $2r$, we prove that at least $\frac{1}{r!}{n}^r\log n-O_r(n^r)$ points are needed to pierce the relative interior of all its codimension-one simplices, which exceeds the number of codimension-one faces in a triangulation by a $\log n$ factor. For equally spaced points on the moment curve in odd dimensions, we construct an optimal blocking set whose size equals the maximum number of codimension-one faces in a triangulation.

12 pages, 1 figure