# Analysis and Probability Seminar: “Empty Simplices”

#### Tuesday March 20, 2018 3:00 p.m. Yost, 306

Speaker: Matthias Reitzner

Title: Empty Simplices

Abstract: Given a finite point set $X$ in the plane, we call a triangle $(x,y,z)$ with vertices in $X$ empty if no other point of $X$ is contained in this triangle.

The degree of a pair $(x,y)$ is the number of empty triangles $(x,y,z)$ with $z \in X$.  Define $\deg X$ as the maximal degree of a pair in $X$. We are interested in the asymptotic behaviour of $\deg X$ if $X$ is a random sample of $n$ independent and uniform points from a fixed convex body.