Unique geodesics for Thompson’s metric

In this paper a geometric characterization of the unique geodesics in Thompson's metric spaces is presented. This characteriza- tion is used to prove a variety of other geometric results. Firstly, it will be shown that there exists a unique Thompson's metric geodesic con- necting x and y in the cone of positive self-adjoint elements in a unital C ∗ -algebra if, and only if, the spectrum of x −1/2 yx −1/2 is contained in {1/�, �} for some � � 1. A similar result will be established for sym- metric 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 ◦ ,dC) can be quasi-isometrically embedded into a finite-dimensional normed space if, and only if, C is a polyhedral cone. Moreover, (C ◦ ,dC) 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 � dimC < 1, then every Thompson's metric isometry is projectively linear.