On Balls in a Hilbert Polygonal Geometry (Multimedia Contribution)

Authors Frank Nielsen, Laetitia Shao

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Frank Nielsen
Laetitia Shao

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Frank Nielsen and Laetitia Shao. On Balls in a Hilbert Polygonal Geometry (Multimedia Contribution). In 33rd International Symposium on Computational Geometry (SoCG 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 77, pp. 67:1-67:4, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)


Hilbert geometry is a metric geometry that extends the hyperbolic Cayley-Klein geometry. In this video, we explain the shape of balls and their properties in a convex polygonal Hilbert geometry. First, we study the combinatorial properties of Hilbert balls, showing that the shapes of Hilbert polygonal balls depend both on the center location and on the complexity of the Hilbert domain but not on their radii. We give an explicit description of the Hilbert ball for any given center and radius. We then study the intersection of two Hilbert balls. In particular, we consider the cases of empty intersection and internal/external tangencies.
  • Projective geometry
  • Hilbert geometry
  • balls


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  1. Curtis T. McMullen. Coxeter groups, Salem numbers and the Hilbert metric. Publications mathématiques de l'IHÉS, 95:151-183, 2002. Google Scholar
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