On the Hilbert Geometry of Convex Polytopes

Abstract

We survey the Hilbert geometry of convex polytopes. In particular we present two important characterisations of these geometries, the first one in terms of the volume growth of their metric balls, the second one as a bi-lipschitz class of the simplexe's geometry.

To appear in the Handbook on Hilbert geometry edited by G. Besson, A. Papadopoulos and M. Troyanov

References in corpus (5)

• On the Hilbert Geometry of Convex Polytopes
• Hilbert geometry for convex polygonal domains
• Spectral Radius and Amenability in Hilbert Geometries
• Lipschitz Characterisation of Polytopal Hilbert Geometries
• Hilbert domains quasi-isometric to normed vector spaces

Cited by in corpus (9)

• The Perron-Frobenius theorem for multi-homogeneous mappings
• On the Hilbert Geometry of Convex Polytopes
• Weighted floating bodies and polytopal approximation
• Hilbert geometry of the Siegel disk: The Siegel-Klein disk model
• Completeness and incompleteness of the Binet-Legendre Metric
• Horofunction compactifications and duality
• Approximability of convex bodies and volume entropy in Hilbert geometry
• Laplacian and spectral gap in regular Hilbert geometries
• Breakdown points of Fermat-Weber problems under gauge distances