The optimal Hölder exponent for max-sliced 1-Wasserstein distance on the unit ball is 2/(d+2), and under a Voronoi matching condition W_p is bounded by a nearly optimal multiple of the sliced distance.
One-Dimensional Empirical Measures, Order Statistics, and
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Nearly sharp comparison results for sliced and max-sliced Wasserstein distances
The optimal Hölder exponent for max-sliced 1-Wasserstein distance on the unit ball is 2/(d+2), and under a Voronoi matching condition W_p is bounded by a nearly optimal multiple of the sliced distance.