Strong equivalence between metrics of Wasserstein type

The sliced Wasserstein metric Wp and more recently max-sliced W?p have attracted abundant attention in data sciences machine learning due to their advantages tackle the curse of dimensionality, see e.g. [15], [6]. A question particular importance is strong equivalence between these projected metrics (classical) Wp. Recently, Paty Cuturi proved [14] W?2 W2. We show that also holds for p=1, while does not share this nice property.