Unique geodesics for Thompson's metric

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 \(x^{-1/2}yx^{-1/2}\) is contained in \(\{1/\beta,\beta\}\) for some \(\beta ≥ 1\). 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 \(3 ≤ dim \space C ≤ \infty\), then every Thompson's metric isometry is projectively linear.