  As the least squares mean for the Riemannian trace metric on the cone of
positive definite matrices, the Riemannian mean with its computational and
theoretical approaches has been widely studied. Recently the new metric and the
least squares mean on the cone of positive definite matrices, which are called
the Wasserstein metric and the Wasserstein mean, respectively, have been
introduced. In this paper, we explore some properties of Wasserstein mean such
as determinantal inequality and find bounds for the Wasserstein mean. Using
bounds for the Wasserstein mean, we verify that the Wasserstein mean is the
multivariate Lie-Trotter mean.
