The Transportation Cost from the Uniform Measure to the Empirical Measure in Dimension ≥ 3

Consider two independent sequences (Xᵢ)i≤ n and (X'ᵢ)i≤ n that are independent and uniformly distributed over 0, 1ᵈ, d ≥ 3. Under mild regularity conditions, we describe the convex functions φ such that, with large probability, there exists a one-to-one map π from \1,…,n\ to \1,…,n\ for which ∑i≤ n1/nφ(Xᵢ - X'π(i)n-1/dK_φ) ≤ 1, where K_φ depends on φ only.