Consider any norm N on Rᵈ, d ≥ 3, and independent uniformly distributed points X₁, …, Xₙ, …; Y₁, …, Yₙ, … in 0, 1ᵈ. Consider the random variable Mₙ = inf ∑i ≤ n N(Xᵢ - Yσ(i)), where the infimum is taken over all permutations σ of \1, …, n\. We show that for some universal constant K, we have lim n → ∞ Mₙ n-1 + 1/d ≤ rN (1 + K log d/d)mathrma,s., where rN is the radius of the ball for N of volume 1.
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Michel Talagrand (1992) studied this question.