Harmonic symmetrization of convex sets and of Finsler structures,with applications to Hilbert geometry |
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Authors: | Athanase Papadopoulos Marc Troyanov |
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Institution: | 1. Institut de Recherche Mathématique Avancée, Université Louis Pasteur and CNRS, 7 rue René Descartes, 67084 Strasbourg Cedex, France;2. Section de Mathématiques, École Polytechnique Fédérale de Lausanne, 1015 Lausanne, Switzerland |
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Abstract: | David Hilbert discovered in 1895 an important metric that is canonically associated to an arbitrary convex domain Ω in the Euclidean (or projective) space. This metric is known to be Finslerian, and the usual proof of this fact assumes a certain degree of smoothness of the boundary of Ω, and refers to a theorem by Busemann and Mayer that produces the norm of a tangent vector from the distance function. In this paper, we develop a new approach for the study of the Hilbert metric where no differentiability is assumed. The approach exhibits the Hilbert metric on a domain as a symmetrization of a natural weak metric, known as the Funk metric. The Funk metric is described as a tautological weak Finsler metric, in which the unit ball in each tangent space is naturally identified with the domain Ω itself. The Hilbert metric is then identified with the reversible tautological weak Finsler structure on Ω, and the unit ball of the Hilbert metric at each point is described as the harmonic symmetrization of the unit ball of the Funk metric. Properties of the Hilbert metric then follow from general properties of harmonic symmetrizations of weak Finsler structures. |
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Keywords: | primary 58B20 secondary 51K05 51K10 52A07 52A20 53B40 53C60 |
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