Consider a geodesic, hyperbolic metric space $(X,d)$. For $x,y \in X$, consider $D(x,y)$ the infimum, over all $1$-Lipschitz paths $p:[a,b] \rightarrow X$ from $x$ to $y$, of the integral $\int_a^b e^{-(x|y)_{p(t)}}dt$, where $(x|y)_{p(t)}$ is the Gromov product.

Is the function $D(x,y)$ metrically proper ? More precisely, does there exist constants $A,B > 0$ such that $D(x,y) \geq Ad(x,y)-B$ ? Has this quantity already been studied ?

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    Do you want $|b - a| = d(x,y)$ ? – Loreno Heer Dec 7 at 23:49
  • No, I do not impose any restrictions on $b-a$. – Thomas Haettel Dec 9 at 12:56
  • So you do not need the 1-Lipschitz condition. Just Lipschitz is enough. Or something I am missing here? – Loreno Heer Dec 9 at 15:08
  • If you parametrize a geodesic from $x$ to $y$ by some interval $[a,b]$ (even Lipschitz), the integral equals $b-a$. So the infimum is zero if you do not restrict to $1$-Lipschitz paths. – Thomas Haettel Dec 9 at 21:33
  • ah, ok I understand – Loreno Heer Dec 9 at 23:01

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