# Wasserstein space with strictly non-positive sectional curvature

Let $$(X,d)$$ be a (separable, complete) metric space with uniformly strictly non-positive curvature in the sense of Alexandrov, i.e. $$(X,d)$$ satisfies a $$CAT(K)$$ inequality for some $$K<0$$.

Does it hold that the 2-Wasserstein space on $$(X,d)$$ has strictly non-positive curvature?

I suspect this is false, but haven't found this result anywhere. Note that the analogous statement for $$CAT(0)$$ is definitely false, because the 2-Wasserstein space on $$\mathbb{R}^d$$ is positively curved (see for example Remark 2.10 in this paper by Bertrand and Kloeckner: https://arxiv.org/pdf/1010.0590.pdf).

Yes, it is false. Note that the space $$W_2(X)$$ contains the space $$\tfrac1{\sqrt{n}}\cdot X^{\times n}/S_n$$ where the group $$S_n$$ permutes the $$X$$-factors. While space $$X^{\times n}$$ is $$\mathrm{CAT}(0)$$, the quotient $$X^{\times n}/S_n$$ is not --- for example, take $$X=$$ Lobachevsky plane.