On the space of positive definite matrices, we consider distance functions of the form 𝑑(𝐴,𝐵)=[tr𝒜(𝐴,𝐵)−tr𝒢(𝐴,𝐵)]1/2,$d(A,B)={[\mathrm{t}\mathrm{r}\mathcal{A}(A,B)-\mathrm{t}\mathrm{r}\mathcal{G}(A,B)]}^{1/2},$ where 𝒜(𝐴,𝐵)$\mathcal{A}(A,B)$ is the arithmetic mean and 𝒢(𝐴,𝐵)$\mathcal{G}(A,B)$ is one of the different versions of the geometric mean. When 𝒢(𝐴,𝐵)=𝐴1/2𝐵1/2$\mathcal{G}(A,B)=A^{1/2}{B}^{1/2}$ this distance is ‖𝐴1/2−𝐵1/2‖2,$\Vert A^{1/2}-B^{1/2}{\Vert }_2,$ and when 𝒢(𝐴,𝐵)=(𝐴1/2𝐵𝐴1/2)1/2$\mathcal{G}(A,B)=(A^{1/2}B{A}^{1/2}{)}^{1/2}$ it is the Bures–Wasserstein metric. We study two other cases: 𝒢(𝐴,𝐵)=𝐴1/2(𝐴−1/2𝐵𝐴−1/2)1/2𝐴1/2,$\mathcal{G}(A,B)=A^{1/2}(A^{-1/2}B{A}^{-1/2}{)}^{1/2}{A}^{1/2},$ the Pusz–Woronowicz geometric mean, and 𝒢(𝐴,𝐵)=exp(log𝐴+log𝐵2),$\mathcal{G}(A,B)=\exp (\frac{\log A+\log B}{2}),$ the log Euclidean mean. With these choices, d(A, B) is no longer a metric, but it turns out that 𝑑2(𝐴,𝐵)$d^2(A,B)$ 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. One of these leads to a new interpretation of a power mean introduced by Lim and Palfia, as a barycentre. The other uncovers interesting relations between the log Euclidean mean and relative entropy.