The horofunction boundary of the Hilbert geometry

By Cormac Walsh. 2006

52 citations recorded by the source corpus.

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.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/812a41598f36e418de0c5b0bc4ab19876c4974a2

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In Applied Mathematics

1. Roger D. Nussbaum
2. Ludovic Marquis
3. Bas Lemmens

In Geometry and Topology

1. Roger D. Nussbaum
2. Ludovic Marquis
3. Bas Lemmens

Citations in the corpus

• Around groups in Hilbert Geometry (2013, background): “The general case: algebraic horospheres Walsh gives a complete description of the horofunction boundary of a Hilbert geometry in [90].”
• On Convex Projective Manifolds and Cusps (2011, unknown): “See Walsh [52] for an extensive discussion.”
• Birkhoff's version of Hilbert's metric and its applications in analysis (2013, background): “The function FC : C ◦ × C → R given by FC(x, y) = logM(x/y) is called the Funk (weak) metric on C, see [56, 68].”