The Euclidean rank of Hilbert geometries

Oliver Bletz-Siebert and Thomas Foertsch

Vol. 231 (2007), No. 2, 257–278

DOI: 10.2140/pjm.2007.231.257

Abstract

We prove that the Euclidean rank of any $3$-dimensional Hilbert geometry $\left(D,h_D\right)$ is 1; that is, $\left(D,h_D\right)$ does not admit an isometric embedding of the Euclidean plane. We show that for higher dimensions this remains true if the boundary $\partial D$ of $D$ is $C^1$.

Keywords

Hilbert geometries, Euclidean rank, asymptotic geometry

Mathematical Subject Classification 2000

Primary: 53C60, 51K99, 51F99

Milestones

Received: 16 March 2006

Revised: 16 November 2006

Accepted: 13 March 2007

Published: 1 June 2007