The horofunction boundary of the Hilbert geometry

Details The horofunction boundary of the Hilbert geometry The horofunction boundary of the Hilbert geometry

2006-11-29

arxiv.org/abs/math/0611920v2

Advances in Geometry, 8 (4) 503-529, 2008

Mathematics

Metric Geometry

24 pages, 2 figures; minor changes, examples added

Scientific paper

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 Painleve-Kuratowski topology.

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