For large N the optimal dispersed subset consists of all points lying outside a self-consistently determined d-dimensional ball, with the entire distribution of maximal dispersion obtained via mean-field order statistics and the replica method.
Mart´ ı et al.,A review on discrete diversity and dispersion maximization from an OR perspective, European Journal of Operational Research299(3), 795-813 (2022)
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The Most Dispersed Subset of Random Points in $\mathbb{R}^d$
For large N the optimal dispersed subset consists of all points lying outside a self-consistently determined d-dimensional ball, with the entire distribution of maximal dispersion obtained via mean-field order statistics and the replica method.