Quantifying minimal non-collinearity among random points

TL;DR

This paper investigates the behavior of the largest angles in triangles formed by random points in a convex set, showing their distribution converges to an exponential form and characterizing the shape that minimizes a certain elongation measure.

Contribution

It establishes the limiting distribution of the largest triangle angle among random points and characterizes the shape minimizing elongation as a ball, including asymptotics for high dimensions.

Findings

Distribution of scaled largest angle converges to exponential as n increases.

The ball minimizes the elongation measure among convex sets.

Asymptotic behavior of elongation in high dimensions is derived.

Abstract

Let $\varphi_{n,K}$ denote the largest angle in all the triangles with vertices among the $n$ points selected at random in a compact convex subset $K$ of $\mathbb{R}^d$ with nonempty interior, where $d\ge2$. It is shown that the distribution of the random variable $\lambda_d(K)\,\frac{n^3}{3!}\,(\pi-\varphi_{n,K})^{d-1}$, where $\lambda_d(K)$ is a certain positive real number which depends only on the dimension $d$ and the shape of $K$, converges to the standard exponential distribution as $n\to\infty$. By using the Steiner symmetrization, it is also shown that $\lambda_d(K)$ -- which is referred to in the paper as the elongation of $K$ -- attains its minimum if and only if $K$ is a ball $B^{(d)}$ in $\mathbb{R}^d$. Finally, the asymptotics of $\lambda_d(B^{(d)})$ for large $d$ is determined.