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Let $X$ be an Hadamard space. For $p\in X$ let $\log_p:X\to T_pX$ be the logarithm map that maps points in $X$ to the corresponding points in the tangent space $T_pX$. Let $μ$ be a Borel probability distribution on $X$ with a second moment. The push-forward distribution $\mu_{\#\log_p}$ on $T_pX$ is defined as $\mu_{\#\log_p}(A)=\mu(\log_p^{-1}(A))$. Let $b(μ)\in X$ the barycenter of $μ$, and $b(\mu_{\#\log_{b(\mu)}})\in T_{b(\mu)}X$ the barycenter of $\mu_{\#\log_{b(\mu)}}$.

Claim: $b(\mu_{\#\log_{b(\mu)}})$ is the cusp of $T_{b(\mu)}X$.

The above claim seems sufficiently simple and useful to be observed before. I'm looking for a reference or references where it is stated and proved.

Thanks in advance.

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    $\begingroup$ I would write that it follows since $(d_x\mathrm{dist}_p^2)(v)=-2\cdot\langle\log_xp,v\rangle$. $\endgroup$ Commented May 3, 2024 at 16:39
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    $\begingroup$ What does "the cusp" mean here? $\endgroup$ Commented May 4, 2024 at 2:25

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