  We offer a new proof of uniform convexity inequalities for the Finsler
manifold of nonpositive curvature taken on the space of positive-semidefinite
matrices with the weighted matrix geometric mean defining the geodesic between
two points. Using the technique of log majorization, we are able to
characterize that the equality cases of said equalities occur if and only if
the matrices commute, and hence are the same as in $\ell^p$.
