Hilbert geometries have bounded local geometry (Q2372835)

A Hilbert geometry is an open, bounded, convex set \(C \) in \(\mathbb R^{n}\) with the distance function defined as follows: For \(p,q\) in \(C \) there are two points \(a,b\) on the line through \(p\) and \(q\) belonging to the boundary of \(C\) with the order \(a,p,q,b\) along the line. Then \(d(p,q)=\frac{1}{2}\ln [a,p,q,b]\) where \([a,p,q,b]\) is the cross ratio. This paper shows that a Hilbert geometry has bounded local geometry, which says that all balls of a given radius are bi-Lipschitz to a Euclidean domain in \(\mathbb R^{n}\). It is also shown that a planar Hilbert geometry which is Gromov hyperbolic has strictly positive Cheeger constant.