A 4/7-limit law for the largest interpoint distance in a rotational ellipsoid

Abstract

Let $M_n$ denote the largest interpoint distance among independent random points $X_1,\dots,X_n$ uniformly distributed in a compact set in $\mathbb{R}^d$. Weak limit laws for $M_n$ are known in several geometric settings, in particular for ellipsoids with a unique major axis. In this paper we treat the simplest nontrivial case in which the largest semi-axis is not unique, namely the rotational ellipsoid $\{(x_1,x_2,x_3)\in\mathbb{R}^3: (x_1^2+x_2^2)/h^2 + x_3^2/a^2 \le 1\}$, where $0<a<h$. The diameter of this ellipsoid is attained by all antipodal pairs on the equatorial circle, so the extremal points are not isolated. We prove that $n^{4/7}(2h-M_n)$ converges in distribution to a Weibull-type limit law with explicit parameter. The proof combines geometric localization arguments with a Chen--Stein Poisson approximation for rare nearly diametral pairs.