And the Funk Metric
نویسنده
چکیده
We discuss general notions of metrics and of Finsler structures which we call weak metrics and weak Finsler structures. Any convex domain carries a canonical weak Finsler structure, which we call its tautological weak Finsler structure. We compute distances in the tautological weak Finsler structure of a domain and we show that these are given by the so-called Funk weak metric. We conclude the paper with a discussion of geodesics, of metric balls and of convexity properties of the Funk weak metric. AMSMathematics Subject Classification: 52A, 53C60, 58B20
منابع مشابه
Funk Metrics and R-Flat Sprays ∗
The well-known Funk metric F (x, y) is projectively flat with constant flag curvature K = −1/4 and the Hilbert metric Fh(x, y) := (F (x, y) + F (x,−y))/2 is projectively flat with constant curvature K = −1. These metrics are the special solutions to Hilbert’s Fourth Problem. In this paper, we construct a non-trivial R-flat spray using the Funk metric. It is then an inverse problem in the calcul...
متن کاملHarmonic Symmetrization of Convex Sets and of Finsler Structures, with Applications to Hilbert Geometry
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 f...
متن کاملThe Funk and Hilbert geometries for spaces of constant curvature
The goal of this paper is to introduce and study analogues of the Euclidean Funk and Hilbert metrics on open convex subsets Ω of hyperbolic or spherical spaces. At least at a formal level, there are striking similarities among the three cases: Euclidean, spherical and hyperbolic. We start by defining non-Euclidean analogues of the Euclidean Funk weak metric and we give three distinct representa...
متن کاملOn Geodesics of Finsler Metrics via Navigation Problem
This paper is devoted to a study of geodesics of Finsler metrics via Zermelo navigation. We give a geometric description of the geodesics of the Finsler metric produced from any Finsler metric and any homothetic field in terms of navigation representation, generalizing a result previously only known in the case of Randers metrics with constant S-curvature. As its application, we present explici...
متن کاملWeak Metrics on Euclidean Domains
A weak metric on a set is a function that satisfies the axioms of a metric except the symmetry and the separation axioms. The aim of this paper is to present some interesting weak metrics and to study some of their properties. In particular, we introduce a weak metric, called the Apollonian weak metric, on any subset of a Euclidean space which is either bounded or whose boundary is unbounded. W...
متن کامل