  In this paper we investigate properties of metric projections onto specific
closed and geodesically convex proper subsets of Wasserstein spaces
$(\mathcal{P}_p(\mathbf{R}^d),W_p).$ When $d=1$, as
$(\mathcal{P}_2(\mathbf{R}),W_2)$ 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 $d>1$, we show the failure of this property in
two regimes: when $p>1$ is either small enough or large enough. Finally, we
prove some positive curvature properties of Wasserstein spaces
$(\mathcal{P}_p(\mathbf{R}^d),W_p)$ when $d\ge 2$ and $p\in(1,+\infty)$ 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.
