Optimal Wp–SWp comparisons for p>1

Establish sharp quantitative inequalities controlling the p‑Wasserstein distance Wp(μ,ν) by the sliced distance SWp(μ,ν) for p>1 on ℝ^d, determining the optimal exponent and its dependence on dimension under natural support or moment assumptions.

Background

Beyond the case p=1 studied in this work, the authors highlight that the regime p>1 is largely unexplored. While one can derive bounds by combining known p=1 estimates with elementary arguments (e.g., yielding an exponent 1/(p(d+1)) for compactly supported measures), such exponents are believed to be far from optimal.

Resolving this would advance the theoretical justification for using sliced distances as proxies for higher‑order Wasserstein metrics in applications, by pinpointing the true quantitative relationship for p>1.