A 4/7-limit law for the largest interpoint distance in a rotational ellipsoid
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866914596571316224 |
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| author | Henze, Norbert |
| author_facet | Henze, Norbert |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_24627 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A 4/7-limit law for the largest interpoint distance in a rotational ellipsoid Henze, Norbert Probability 60D05, 60F05, 60G70 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. |
| title | A 4/7-limit law for the largest interpoint distance in a rotational ellipsoid |
| topic | Probability 60D05, 60F05, 60G70 |
| url | https://arxiv.org/abs/2605.24627 |