Unique geodesics for Thompson's metric

2013/05/28 by Lemmens, Bas, Roelands, Mark · 1 citation
#53C22 #FOS: Mathematics #Functional Analysis (math.FA) #Metric Geometry (math.MG)

paper · doi:10.48550/arxiv.1305.6643

Abstract

In this paper a geometric characterization of the unique geodesics in Thompson's metric spaces is presented. This characterization is used to prove a variety of other geometric results. Firstly, it will be shown that there exists a unique Thompson's metric geodesic connecting x and y in the cone of positive self-adjoint elements in a unital C^*-algebra if, and only if, the spectrum of x^-1/2yx^-1/2 is contained in \1/β,β\ for some β≥ 1. A similar result will be established for symmetric cones. Secondly, it will be shown that if C^∘ is the interior of a finite-dimensional closed cone C, then the Thompson's metric space (C^∘,d_C) can be quasi-isometrically embedded into a finite-dimensional normed space if, and only if, C is a polyhedral cone. Moreover, (C^∘,d_C) is isometric to a finite-dimensional normed space if, and only if, C is a simplicial cone. It will also be shown that if C^∘ is the interior of a strictly convex cone C with 3≤ dim C

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