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Advances in Geometry

Managing Editor: Grundhöfer, Theo / Joswig, Michael

Editorial Board: Bamberg, John / Bannai, Eiichi / Cavalieri, Renzo / Coskun, Izzet / Duzaar, Frank / Eberlein, Patrick / Gentili, Graziano / Henk, Martin / Kantor, William M. / Korchmaros, Gabor / Kreuzer, Alexander / Lagarias, Jeffrey C. / Leistner, Thomas / Löwen, Rainer / Ono, Kaoru / Ratcliffe, John G. / Scheiderer, Claus / Van Maldeghem, Hendrik / Weintraub, Steven H. / Weiss, Richard

IMPACT FACTOR 2017: 0.734

CiteScore 2017: 0.70

SCImago Journal Rank (SJR) 2017: 0.695
Source Normalized Impact per Paper (SNIP) 2017: 0.891

Mathematical Citation Quotient (MCQ) 2017: 0.62

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Volume 8, Issue 4


The horofunction boundary of the Hilbert geometry

Cormac Walsh
Published Online: 2008-11-28 | DOI: https://doi.org/10.1515/ADVGEOM.2008.032


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

About the article

Received: 2006-12-21

Revised: 2007-11-20

Published Online: 2008-11-28

Published in Print: 2008-10-01

Citation Information: Advances in Geometry, Volume 8, Issue 4, Pages 503–529, ISSN (Online) 1615-7168, ISSN (Print) 1615-715X, DOI: https://doi.org/10.1515/ADVGEOM.2008.032.

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