Hilbert geometry of polytopes

Bernig, Andreas

doi:10.1007/s00013-009-3142-1

Cited by 18 publications

(11 citation statements)

References 12 publications

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“…Analogous results for Hilbert's metric spaces were obtained by Colbois and Verovic [6], and by Foertsch and Karlsson [12], see also [3]. Our method of proof is similar to theirs, but interesting adaptations need to be made to make the arguments work.…”

Section: Annales De L'institut Fouriersupporting

confidence: 57%

Unique geodesics for Thompson’s metric

Lemmens¹,

Roelands²

2015

Annales De l'Institut Fourier

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Section: Annales De L'institut Fouriersupporting

confidence: 57%

Unique geodesics for Thompson’s metric

Lemmens¹,

Roelands²

2015

Annales De l'Institut Fourier

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“…Theorem [Bernig 2009;Vernicos 2008b]. The Hilbert metric associated to a convex body K is bi-Lipschitz to a normed space if and only if K is a polytope.…”

Section: Introductionmentioning

confidence: 99%

Volume entropy of Hilbert geometries

Berck¹,

Bernig²,

Vernicos³

2010

Pacific J. Math.

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We show that among all plane Hilbert geometries, the hyperbolic plane has maximal volume entropy. More precisely, we show that the volume entropy is bounded above by 2/(3 − d) ≤ 1, where d is the Minkowski dimension of the extremal set of K , and we construct an explicit example of a plane Hilbert geometry with noninteger volume entropy. In arbitrary dimension, the hyperbolic space has maximal entropy among all Hilbert geometries satisfying some additional technical hypothesis. To achieve this result, we construct a new projective invariant of convex bodies, similar to the centroaffine area.

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“…This theorem was proved by Bruno Colbois, Patrick Verovic and Constantin Vernicos in dimension 2 [13], and independently by Andreas Bernig [8] and by the author [29] in all dimensions.…”

Section: Polytopal Hilbert Geometries Are Bi-lipschitz Tomentioning

confidence: 97%