Quantifying minimal noncollinearity among random points

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 $\mathbf {R}^d$. Finally, the asymptotics of $\lambda_d(B^{(d)})$ for large $d$ is determined.