A Note on Empty Balanced Tetrahedra in Two colored Point sets in $\mathbb{R}^3$

Abstract

Let $S$ be a set of $n$ red and $n$ blue points in general position in $\mathbb{R}^3$. Let $\tau$ be a tetrahedra with vertices on $S$. We say that $\tau$ is \emph{empty} if it does not contain any point of $S$ in its interior. We say that $\tau$ is \emph{balanced} if it contains two blue vertices and two red vertices. In this paper we show that $S$ spans $\Omega(n^{5/2})$ empty balanced tetrahedra.