Upper bounds for stabbing simplices by a line

TL;DR

This paper investigates upper bounds for line stabbing simplices in high-dimensional point sets, comparing two specific configurations and providing numerical bounds for dimensions 4, 5, and 6.

Contribution

It introduces a detailed analysis of upper bounds for line stabbing simplices using the stretched grid and diagonal configurations, with new numerical bounds for dimensions 4 to 6.

Findings

Stretched grid yields better bounds than stretched diagonal for dimensions 4, 5, 6.

Numerical bounds for $c_{d,1}$ are provided for $d=4,5,6$.

Calculations involve complex analytical and numerical methods.

Abstract

It is known that for every dimension $d\ge 2$ and every $k<d$ there exists a constant $c_{d,k}>0$ such that for every $n$-point set $X\subset \mathbb R^d$ there exists a $k$-flat that intersects at least $c_{d,k} n^{d+1-k} - o(n^{d+1-k})$ of the $(d-k)$-dimensional simplices spanned by $X$. However, the optimal values of the constants $c_{d,k}$ are mostly unknown. The case $k=0$ (stabbing by a point) has received a great deal of attention. In this paper we focus on the case $k=1$ (stabbing by a line). Specifically, we try to determine the upper bounds yielded by two point sets, known as the "stretched grid" and the "stretched diagonal". Even though the calculations are independent of $n$, they are still very complicated, so we resort to analytical and numerical software methods. We provide strong evidence that, surprisingly, for $d=4,5,6$ the stretched grid yields better bounds than the stretched diagonal (unlike for all cases $k=0$ and for the case $(d,k)=(3,1)$, in which both point sets yield the same bound). Our experiments indicate that the stretched grid yields $c_{4,1}\leq 0.00457936$, $c_{5,1}\leq 0.000405335$, and $c_{6,1}\leq 0.0000291323$.