Hilbert Geometry for Strictly Convex Domains

AbstractWe prove in this paper that the Hilbert geometry associated with a bounded open convex domain $$\mathcal{C}$$ in Rn whose boundary $$\partial \mathcal{C}$$ is a $$\mathcal{C}$$ 2 hypersuface with nonvanishing Gaussian curvature is bi-Lipschitz equivalent to the n-dimensional hyperbolic space Hn. Moreover, we show that the balls in such a Hilbert geometry have the same volume growth entropy as those in Hn.