Empty Monochromatic Simplices

TL;DR

This paper investigates the number of empty monochromatic simplices in multi-colored point sets in high-dimensional space, providing new lower bounds and triangulation results that advance understanding of geometric combinatorics.

Contribution

It establishes new lower bounds on the count of empty monochromatic simplices and proves the existence of triangulations with many simplices in high dimensions.

Findings

Lower bound of (n^{d-k+1+2^{-d}}) for 3  k  d

Improved lower bound of (n^{d-2/3}) for k=2

Existence of triangulations with at least dn + (\,log n) simplices in (d) dimensions

Abstract

Let $S$ be a $k$-colored (finite) set of $n$ points in $\mathbb{R}^d$, $d\geq 3$, in general position, that is, no {$(d + 1)$} points of $S$ lie in a common $(d - 1)$}-dimensional hyperplane. We count the number of empty monochromatic $d$-simplices determined by $S$, that is, simplices which have only points from one color class of $S$ as vertices and no points of $S$ in their interior. For $3 \leq k \leq d$ we provide a lower bound of $\Omega(n^{d-k+1+2^{-d}})$ and strengthen this to $\Omega(n^{d-2/3})$ for $k=2$. On the way we provide various results on triangulations of point sets in $\mathbb{R}^d$. In particular, for any constant dimension $d\geq3$, we prove that every set of $n$ points ($n$ sufficiently large), in general position in $\mathbb{R}^d$, admits a triangulation with at least $dn+\Omega(\log n)$ simplices.