  On the space of positive definite matrices we consider distance functions of
the form $d(A,B)=\left[\tr\mathcal{A}(A,B)-\tr\mathcal{G}(A,B)\right]^{1/2},$
where $\mathcal{A}(A,B)$ is the arithmetic mean and $\mathcal{G}(A,B)$ is one
of the different versions of the geometric mean. When
$\mathcal{G}(A,B)=A^{1/2}B^{1/2}$ this distance is $\|A^{1/2}-B^{1/2}\|_2,$ and
when $\mathcal{G}(A,B)=(A^{1/2}BA^{1/2})^{1/2}$ it is the Bures-Wasserstein
metric. We study two other cases:
$\mathcal{G}(A,B)=A^{1/2}(A^{-1/2}BA^{-1/2})^{1/2}A^{1/2},$ the Pusz-Woronowicz
geometric mean, and $\mathcal{G}(A,B)=\exp\big(\frac{\log A+\log B}{2}\big),$
the log Euclidean mean. With these choices $d(A,B)$ is no longer a metric, but
it turns out that $d^2(A,B)$ is a divergence. We establish some (strict)
convexity properties of these divergences. We obtain characterisations of
barycentres of $m$ positive definite matrices with respect to these distance
measures.
