Embedding $X \ni x \mapsto \delta_x \in P_2(X)$ is totally convex

Question

I am looking for a reference to a proof of the following result:

Let $X$ be a compact, connected, smooth Riemannian manifold. Then, the embedding $$X \ni x \mapsto \delta_x \in P_2(X) $$ has totally convex image.

In the above, $P_2(X)$ denotes the $2$-Wasserstein space on $X$, and $\delta_x$ is the Dirac measure at $x$.

I saw this result in Lott and Villani's paper on synthetic Ricci curvature bounds. In Villani's books on optimal transport, I was able to find a proof that the above map is an isometric embedding, but not that it has totally convex image.

Thank you in advance for the reference.

Answer 1

The statement is false, as @Del pointed out, and the same example can be found here: https://arxiv.org/abs/1105.2883 (the paragraph after Proposition 3.7; be careful with the terminology in this paper and what I say in the next paragraph!).

Lott's and Villani's paper mentions that the map is totally geodesic, which, in the context of, for instance, proving that the $2$-Wasserstein space has non-negative curvature iff $X$ does, I think means that geodesics in $X$ are mapped to geodesics in $P_2(X)$.