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Let $(X,d)$ be a complete locally-compact metric space. We define the $p$-barycenter map as a continuous function: $$ \beta:\mathcal{P}_p(X)\rightarrow X, $$ which is a right-inverse of the map associating to any $x \in X$ its point-mass $\delta_x\in \mathcal{P}_p(X)$.

It is well-known that, if $X$ is CAT(0) then $\beta$ can be assumed to be $L$-Lipschitz for some $L\in (0,1]$. My question is, are there known (resp. classes of) examples of metric spaces for which $\beta$ is Hölder continuous but not Lipschitz continuous?

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  • $\begingroup$ What is the variable $p$ supposed to denote? $\endgroup$
    – YCor
    Commented Jul 13, 2021 at 9:51
  • $\begingroup$ Oh, it denotes the $p^{th}$ Wasserstein space; i.e. $\mu\in \mathcal{P}(X)$ for which $\int_{x \in X} d^p(x,x_0)\mu(dx)<\infty$ for some $x_0\in X$ and with the usual $p$-Wasserstein distance. $\endgroup$
    – ABIM
    Commented Jul 13, 2021 at 10:07

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