When are Wasserstein spaces $CAT(\kappa)$?

Let $$(X,d)$$ be a complete and separable metric space and, for $$1\leq p<\infty$$, let $$(\mathcal{P}_p(X,d),W_p)$$ be the $$p$$-Wasserstein space on $$(X,d)$$. For which $$p$$ and $$(X,d)$$ is $$(\mathcal{P}_p(X,d),W_p)$$ a $$CAT(\kappa)$$ space?

I know that for $$p=2$$ and $$(X,d)$$ a Banach space, $$(\mathcal{P}_2(X,d),W_2)$$ is a Hadamard ($$CAT(0)$$) space; but there must be other cases...

• Your statement "I know that..." might hold ONLY for Hilbert spaces! Dec 5, 2021 at 4:11
• @AntonPetrunin Is it even true for Hilbert spaces? I can't actually see why (or find a reference) Feb 8, 2022 at 12:08

Note that there is an isometric embedding $$X\to W_p(X)$$, so $$X$$ has to be CAT(κ). Second the space $$W_p(X)$$ contains symmetric $$p$$-product $$S^n(X)=X^{\times n}/S_n$$ so $$p=2$$, or $$X$$ is one a point-space. Now if $$\dim X>1$$, then you get into trouble with extending geodesic thru a $$\delta$$-measure in $$S^2(X)$$, so you get $$\dim X\le1$$.
• I was thinking, is $\mathcal{W}_p(X)$ ever locally $CAT(\kappa)$? Or does something rule that out also? Dec 20, 2021 at 23:42
• $X$ could be one-point space, or $\mathbb{R}$, maybe a tree. Dec 21, 2021 at 7:48