Abstract
Recently Hiai-Petz [7] introduced two types of interesting geometries of
positive-definite matrices whose geodesics are paths of operator means and then the
author [5] showed these geometries have Finsler structures for all unitarily invariant
norms. Though the geodesic is of the shortest length between fixed two matrices,
the shortest paths are not unique in general as pointed out in [7]. In this paper, we
show that their geodesic is the unique shortest path in each Hiai-Petz geometry for
all strongly convex unitarily invariant norms. As counter examples, we show that this
uniqueness is false for Ky Fan norms.
