# Integral of Gromov product

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 ?

• 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