Entropy degeneration of convex projective surfaces

We show that the volume entropy of the Hilbert metric on a closed convex projective surface tends to zero as the corresponding Pick differential tends to infinity. The proof is based on the theorem, due to Benoist and Hulin, that the Hilbert metric and Blaschke metric are comparable.Comment: 5 page

Hilbert metric in the unit ball

The Hilbert metric between two points $x,y$ in a bounded convex domain $G$ is defined as the logarithm of the cross-ratio of $x,y$ and the intersection points of the Euclidean line passing through the points $x,y$ and the boundary of the domain. Here, we study this metric in the case of the unit ball $\mathbb{B}^n$. We present an identity between the Hilbert metric and the hyperbolic metric, give several inequalities for the Hilbert metric, and results related to the inclusion properties of the balls defined in the Hilbert metric. Furthermore, we study the distortion of the Hilbert metric under conformal mappings.Comment: 16 pages, 4 figure

Hilbert geometry of polytopes

It is shown that the Hilbert metric on the interior of a convex polytope is bilipschitz to a normed vector space of the same dimension.Comment: 11 pages, minor changes, to appear in Archiv Mat

Hilbert metric, beyond convexity

The Hilbert metric on convex subsets of $\mathbb R^n$ has proven a rich notion and has been extensively studied. We propose here a generalization of this metric to subset of complex projective spaces and give examples of applications to diverse fields. Basic examples include the classical Hilbert metric which coincides with the hyperbolic metric on real hyperbolic spaces as well as the complex hyperbolic metric on complex hyperbolic spaces

Software and Analysis for Dynamic Voronoi Diagrams in the Hilbert Metric

The Hilbert metric is a projective metric defined on a convex body which generalizes the Cayley-Klein model of hyperbolic geometry to any convex set. In this paper we analyze Hilbert Voronoi diagrams in the Dynamic setting. In addition we introduce dynamic visualization software for Voronoi diagrams in the Hilbert metric on user specified convex polygons