Geometry of midpoint sets for Thompson’s metric

It is well-known that the convex cone Pm of m×m positive definite matrices is a Cartan–Hadamard Riemannian manifold with respect to the Riemannian trace metric where the geometric mean of two positive definite matrices coincides with the unique metric midpoint between them. In this paper we consider the Thompson metric on Pm inherited from the spectral norm and study some geometric structures of the Thompson midpoints. We prove that there is a unique midpoint (minimal geodesic) between A and B if and only if the spectrum of A-1B is contained in {a,a-1} for some a>0, and the set of Thompson midpoints between A and B is compact and is convex in both Riemannian and Euclidean sense. It is further shown that the set of all weighted midpoints between A and B is compact and convex in Riemannian sense.