Linearity of Cartan and Wasserstein means

The convex cone of positive definite Hermitian matrices has two important Riemannian geometries, where the parametrized weighted geometric (alternatively, Cartan) and Wasserstein means appear as the corresponding geodesics. The major problem with which this paper is concerned is linearity of Cartan and Wasserstein geodesics; when the Cartan (resp. Wasserstein) geodesic between two positive definite matrices A and B does lie in the space spanned by them and what the path in the plane to which it corresponds is. We settle the problem completely for the Cartan geometry and partially for the Wasserstein geometry.