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...

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

1 Answer 1


Almost never...

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 $\dim X\le1$.

  • $\begingroup$ I was thinking, is $\mathcal{W}_p(X)$ ever locally $CAT(\kappa)$? Or does something rule that out also? $\endgroup$ Dec 20, 2021 at 23:42
  • $\begingroup$ $X$ could be one-point space, or $\mathbb{R}$, maybe a tree. $\endgroup$ Dec 21, 2021 at 7:48

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