*Submitted Paper*

**Inserted:** 29 aug 2020

**Last Updated:** 6 sep 2020

**Year:** 2020

**Abstract:**

In this note we investigate properties of metric projection operators onto closed and geodesically convex proper subsets of Wasserstein spaces $(\mathcal{P}_p(\mathbb{R}^d),W_p).$ In our study we focus on the particular subset of probability measures having densities uniformly bounded by a given constant. When $d=1$, $(\mathcal{P}_2(\mathbb{R}),W_2)$ is isometrically isomorphic to a flat space with a Hilbertian structure, and so the corresponding projection operators are nonexpansive. We prove a general ``weak nonexpansiveness" property in arbitrary dimension which provides (among other things) a direct proof of nonexpansiveness when $d=1$. When $d>1$, the space $(\mathcal{P}_2(\mathbb{R}^d),W_2)$ is non-negatively curved in the sense of Alexandrov. So, the question of nonexpansiveness of projection operators on $(\mathcal{P}_p(\mathbb{R}^d),W_p)$ is more subtle. We show the failure of this property for $p\in (1,p(d))$ and we give a quantitative asymptotic estimate on $p(d)>1$ as $d \to \infty$. This result heuristically provides an argument for the fact that $(\mathcal{P}_p(\mathbb{R}^d),W_p)$ is non-negatively curved if $p\in (1,p(d))$. Further geometric properties of independent interest are also discussed.

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