Sharpness of the 2/(d+2) exponent in the W1–MSW1 comparison

Determine whether the exponent 2/(d+2) in the inequality comparing the 1‑Wasserstein distance W1(μ,ν) and the max‑sliced distance MSW1(μ,ν) for probability measures supported on a ball in ℝ^d, namely W1(μ,ν) ≲ R^{d/(d+2)} MSW1(μ,ν)^{2/(d+2)}, is optimal. Precisely, prove or refute that the exponent 2/(d+2) is sharp in general (beyond the known asymptotic lower bound 2/d).

Background

The paper reviews recent progress on quantitative comparisons between classical Wasserstein distances and sliced variants. For MSW1, Bobkov and Götze established an inequality of the form W1(μ,ν) ≲ R^{d/(d+2)} MSW1(μ,ν)^{2/(d+2)} for measures supported in a ball, improving earlier results by Hahn and Quinto. They also showed that the exponent cannot be better than 2/d (asymptotically in d).

The authors note that while the inequality is established, whether the exponent 2/(d+2) is the exact optimal exponent remains unresolved. Clarifying this would settle the precise strength of max‑sliced distances as surrogates for W1 in high dimensions.