The horofunction boundary of the Hilbert geometry

Abstract

We investigate the horofunction boundary of the Hilbert geometry defined on an arbitrary finite-dimensional bounded convex domain D. We determine its set of Busemann points, which are those points that are the limits of `almost-geodesics'. In addition, we show that any sequence of points converging to a point in the horofunction boundary also converges in the usual sense to a point in the Euclidean boundary of D. We prove that all horofunctions are Busemann points if and only if the set of extreme sets of the polar of D is closed in the Painleve-Kuratowski topology.

24 pages, 2 figures; minor changes, examples added

References in corpus (1)

• The horofunction boundary of finite-dimensional normed spaces

Cited by in corpus (6)

• A maximin characterization of the escape rate of nonexpansive mappings in metrically convex spaces
• The horofunction boundary of a Gromov hyperbolic space
• Gauge-reversing maps on cones, and Hilbert and Thompson isometries
• Horofunction compactifications and duality
• A metric version of Poincaré's theorem concerning biholomorphic inequivalence of domains
• Firm non-expansive mappings in weak metric spaces