Setup:
Let $X$ be a complete pointed metric space.
Let us briefly recall that the Wasserstein space $W_1(X)$ is identifiable with a subset of the Arens-Eells (or Lipschitz-Free) space $\operatorname{AE}(X)$ (i.e.: the pre-dual of $\operatorname{Lip}_0(X)$), (via the identification of the (dense) set of finitely-supported probability measures $\{\sum_{i=1}^I k_i\delta_{x_i}\}\subseteq W_1(X)$ with its corresponding elements $\{\sum_{i=1}^I k_iE_{x_i}\}\subseteq W_1(X)$ in the pre-dual of $\operatorname{Lip}_0(X)$); where $E_x:\operatorname{Lip}_0(X):\ni f \mapsto f(x)\in \mathbb{R}$ is the point-evaluation map.
I'm having trouble relating the following two, "seemingly interchangeable" notions.
(B1) In the theory of Lipschitz-spaces: a barycenter map is a continuous left-inverse of the map $\delta:X\rightarrow \operatorname{AE}(X)$ sending $x$ to the point-evaluation map $x\mapsto E_x\in \operatorname{AE}(X)$.
(B2) In Optimal Transport/Probability Theory: a barycentre map refers to a continuous left-inverse of the map $x\mapsto \delta_x\in W_1(X)$.
Question:
Clearly, via the isometric identification of $W_1(X)$ with a subset of $\operatorname{AE}(X)$, if (B1) holds, then (B2) must also. However, is the converse true? That is, if (B2) holds, then can the map $x\mapsto \delta_x$ be extended to all of $\operatorname{AE}(X)$?
Esp. when this map is Lipschitz, then can the extension also be concluded to be Lipschitz?
