The limit law of the largest interpoint distance in a $d$-dimensional ellipsoid

Abstract: We consider the largest interpoint distance $M_n=\max_{1\le i&lt;j\le n}|X_i-X_j|$ among independent random points $X_1,\ldots,X_n$, uniformly distributed on a $d$-dimensional ellipsoid. We assume that the largest semi-axis has length 1 and multiplicity $k\ge 2$, whereas the remaining semi-axes are strictly smaller. In this situation, the diameter is attained on a manifold of dimension $k-1$, and the extremal points are no longer isolated. We establish a weak limit law for the diameter deficit $2-M_n$. Writing $q=d-k$ and $α=q+(k+3)/2$, we show that $n<sup>{2/α}(2-M_n)$ converges in distribution to a Weibull random variable. The proof is based on a local analysis near the diameter manifold, a sharp asymptotic formula for the two-point tail probability, and a Chen--Stein Poisson approximation for rare nearly diametral pairs.

• The paper establishes that the rescaled diameter deficit n^(2/α)(2 − Mₙ) converges to a Weibull distribution.
• It derives the normalization exponent 2/α, where α is computed from the ellipsoid's contracting axes and the multiplicity of its major axis.
• The work employs local geometric analysis and Poisson approximation techniques to address non-isolated extremal points on the diameter manifold.

Limit Law for the Largest Interpoint Distance in a $d$-Dimensional Ellipsoid

Problem Context and Motivation

The paper addresses the asymptotic behavior of the largest Euclidean distance, $M_n = \max_{i<j} \|X_i - X_j\|$, among $n$ independent, uniformly distributed random points inside a $d$-dimensional ellipsoid $E$. The central question is to establish the probabilistic limits for how close $M_n$ is to the geometric diameter of $E$ as $n \to \infty$, together with precise normalization rates and limiting distributions.

Distinctively, the work considers ellipsoids whose largest semi-axis has multiplicity $k\ge2$, while the remainder are strictly smaller. The diameter is thus realized not at isolated points, but on a $(k-1)$-dimensional manifold. This stands in contrast to traditional settings focusing on bodies with isolated extremal points, such as ellipsoids with a unique major axis. The non-isolated nature of the extremal set fundamentally alters the geometric and probabilistic analysis required.

Main Results

The principal contribution is a characterization of the weak limit law for the "diameter deficit" $M_n = \max_{i<j} \|X_i - X_j\|$0, that is, the asymptotic behavior of the random shortfall between $M_n = \max_{i<j} \|X_i - X_j\|$1 and the diameter ($M_n = \max_{i<j} \|X_i - X_j\|$2) of the ellipsoid. The key findings are as follows:

• Let $M_n = \max_{i<j} \|X_i - X_j\|$3 be the multiplicity of the largest semi-axis, $M_n = \max_{i<j} \|X_i - X_j\|$4 the number of contracting axes, and define the effective dimension parameter $M_n = \max_{i<j} \|X_i - X_j\|$5.
• The rescaled diameter deficit $M_n = \max_{i<j} \|X_i - X_j\|$6 converges in distribution to a Weibull random variable.
• The explicit limiting distribution function is $M_n = \max_{i<j} \|X_i - X_j\|$7, where $M_n = \max_{i<j} \|X_i - X_j\|$8 is a computable positive constant dependent on the local geometry near the diameter manifold.

This result generalizes prior work (e.g., [he26], [sc16]) by encompassing arbitrary $M_n = \max_{i<j} \|X_i - X_j\|$9 and $n$0 and thus covers a broader class of ellipsoidal supports where the extremal set is a compact submanifold.

Technical Approach

The analysis harnesses advanced techniques from geometric probability, local asymptotic analysis, and Poisson approximation methods for rare events. The major steps are:

• Local Geometric Analysis: The ellipsoid's structure is exploited to introduce local coordinates adapted to the diameter manifold $n$1. Points with pairwise distance close to $n$2 must be near this manifold, with their $n$3-dimensional components nearly antipodal and the remaining $n$4 coordinates near zero.
• Asymptotic Expansion: Through precise Taylor expansion near the diameter manifold, the diameter deficit is represented as a sum of quadratic forms in the local coordinates, up to an asymptotically negligible remainder.
• Localization Lemma: It is shown that any almost-diametral pair must be concentrated in a vanishingly small neighborhood of the diameter manifold as $n$5.
• Two-Point Tail Probability: The probability that any randomly selected pair is within $n$6 of the diameter is asymptotically proportional to $n$7. This is made explicit via a sharp asymptotic integral, capturing the manifold measure and local density.
• Poisson (Chen–Stein) Approximation: The event of observing at least one nearly diametral pair among all $n$8 possibilities is shown to obey Poisson statistics. This is justified by bounding higher-dependence probabilities using combinatorial and geometric volume estimates.
• Limit Theorem Derivation: The full distributional limit for the suitably rescaled $n$9 is deduced from the convergence of the associated Poisson process, yielding the Weibull form.

Numerical and Theoretical Implications

A particularly notable consequence is the establishment of the normalization exponent $d$0, which depends delicately on both the dimension of the diameter manifold and the number of contracting axes. For example:

• For $d$1, $d$2, $d$3, one recovers $d$4 and normalization $d$5, as in earlier results for rotational ellipsoids [he26].
• For $d$6 (unique major axis, isolated extremal points), $d$7, aligning with the normalization in [sc16].

The approach further demonstrates that in non-spherically symmetric bodies with nontrivial diameter manifolds, extreme-value statistics cannot be reduced to classical extreme value distributions but, instead, require geometric and measure-theoretic integration over the extremal manifold.

The paper's techniques and findings have potential impact on the analysis of high-dimensional data, random geometric graphs, and statistical inference on random point clouds, particularly in disciplines such as spatial statistics, astronomy, and multivariate extremes.

Directions for Future Work

If the uniform distribution on the ellipsoid is replaced by a more general density that is continuous and positive near the diameter manifold, it is conjectured that the normalization exponent $d$8 remains unchanged, though the limiting constant will depend on the local density.

Further generalization to other smooth convex bodies or regions where the diameter is attained on more intricate manifolds is proposed as an open challenge. The precise interplay between the dimension of the extremal set and the local behavior of the boundary remains partially unexplored. Extension of these ideas to broader classes of compact supports and applications in, e.g., directional statistics, represents promising directions.

Conclusion

The paper provides a rigorous and comprehensive treatment of the largest interpoint distance among uniformly distributed random points in an ellipsoid with a high-dimensional manifold of diameter points. The limiting Weibull law and explicit normalization reflect the complex interplay between global geometry and local manifold structure. The methodology paves the way for further advances in geometric probability and the theory of extremes for random structures with manifold-type extremal sets (2606.29036).