Abstract In this paper, we investigate properties of metric projections onto specific closed and geodesically convex proper subsets of Wasserstein spaces (P p (R d), W p) (P (R^d), W). When d = 1 d=1, as (P 2 (R), W 2) (P₂ (R), W₂) is isometrically isomorphic to a flat space with a Hilbertian structure, the corresponding projection operators are expected to be nonexpansive. We give a direct proof of this fact, relying on intrinsic analysis, which also implies nonexpansiveness in certain special cases in higher dimensions. When d > 1 d>1, we show the failure of this property in two regimes: when p > 1 p>1 is either small enough or large enough. Finally, we prove some positive curvature properties of Wasserstein spaces (P p (R d), W p) (P (R^d), W) when d ≥ 2 d 2 and p ∈ (1, + ∞) p (1, +) are arbitrary: we show that Wasserstein spaces are nowhere locally Busemann NPC spaces, and they nowhere locally satisfy the so-called projection criterion. As a corollary of the former, they have nonnegative upper Alexandrov curvature, in a precise sense that we define here. In our analysis, a particular subset of probability measures having densities uniformly bounded above by a given constant plays a special role.
Adve et al. (Tue,) studied this question.
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