Hilbert geometry for strictly convex domains (Q1827414)

Given a bounded open convex body \(C\) in the Euclidean space \((\mathbb R^n,\| \cdot\| )\) the authors investigate the properties of the Hilbert metric \(d_C(x,y)=\frac12\ln \big(\frac{\| y-a\| }{\| y-b\| }:\frac{\| x-a\| }{\| x-b\| }\big)\) where \(a,b\) are the meeting points of the line \((x,y)\) and the boundary \(\partial C\) of \(C\). If \(C\) is the unit ball, then \((C,d_C)\) is the Klein model of the \(n\)-dimensional hyperbolic space \(\mathbb H^n\). In Theorem 2.2 the authors prove that the metric space \((C,d_C)\) is bi-Lipschitz equivalent to the hyperbolic space \(\mathbb H^n\) if \(C\) is strictly convex in the sense that the boundary \(\partial C\) is a \(C^2\) hypersurface with non-vanishing Gaussian curvature. Also, for such a set \(C\) the volume \(\mu_C(B(c,t))\) of the ball \(B(c,t)\) of radius \(t\) centered at a fixed point \(c\in C\) has exponential growth. More precisely, \(\lim_{t\to\infty}\frac{\ln[\mu_C(B(c,t))]}t=n-1\) according to Theorem 3.1.