Quantifying minimal non-collinearity among random points

Abstract

Let ${\phi }_{n,K}$ 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 $d\ge 2$. It is shown that the distribution of the random variable ${\lambda }_d(K)\frac{n^3}{3!}(\pi -{\phi }_{n,K}{)}^{d-1}$, 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.

7 pages. Version 2: minor mistakes and typos are fixed. Version 3: it is also shown that the elongation attains its minimum only on balls, and the large-dimension asymptotics of the elongation for balls is determined