  In this paper we study the monotonicity, in-betweenness and in-sphere
properties of matrix means with respect to Bures-Wasserstein, Hellinger and
Log-Determinant metrics. More precisely, we show that the matrix power means
(Kubo-Ando and non-Kubo-Ando extensions) satisfy the in-betweenness property in
the Hellinger metric. We also show that for two positive definite matrices $A$
and $B$, the curve of weighted Heron means, the geodesic curve of the
arithmetic and the geometric mean lie inside the sphere centered at the
geometric mean with the radius equal to half of the Log-Determinant distance
between $A$ and $B$.
