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The horofunction boundary of the Hilbert geometry
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Cormac Walsh
Published/Copyright:
November 28, 2008
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 Painlevé–Kuratowski topology.
Key words.: Hilbert geometry; Hilbert's projective metric; horoball; max-plus algebra; metric boundary; Busemann function
Received: 2006-12-21
Revised: 2007-11-20
Published Online: 2008-11-28
Published in Print: 2008-October
© de Gruyter 2008
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- Finite type Monge–Ampère foliations
- The horofunction boundary of the Hilbert geometry
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Keywords for this article
Hilbert geometry;
Hilbert's projective metric;
horoball;
max-plus algebra;
metric boundary;
Busemann function
Articles in the same Issue
- Homogeneous geodesics of non-unimodular Lorentzian Lie groups and naturally reductive Lorentzian spaces in dimension three
- Finite type Monge–Ampère foliations
- The horofunction boundary of the Hilbert geometry
- Counting the hyperplane sections with fixed invariants of a plane quintic – three approaches to a classical enumerative problem
- Totally non-real divisors in linear systems on smooth real curves
- Covers of Klein surfaces
- A classification of polarized manifolds by the sectional Betti number and the sectional Hodge number
- On reduced polytopes and antipodality
