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,\documentclass[12pt]{minimal}
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\begin{document}$$d(A,B)=\left[ \mathrm{tr}\mathcal {A}(A,B)-\mathrm{tr}\mathcal {G}(A,B)\right] ^{1/2},$$\end{document} where A(A,B)\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {A}(A,B)$$\end{document} is the arithmetic mean and G(A,B)\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {G}(A,B)$$\end{document} is one of the different versions of the geometric mean. When G(A,B)=A1/2B1/2\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {G}(A,B)=A^{1/2}B^{1/2}$$\end{document} this distance is ‖A1/2-B1/2‖2,\documentclass[12pt]{minimal}
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\begin{document}$$\Vert A^{1/2}-B^{1/2}\Vert _2,$$\end{document} and when G(A,B)=(A1/2BA1/2)1/2\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {G}(A,B)=(A^{1/2}BA^{1/2})^{1/2}$$\end{document} it is the Bures–Wasserstein metric. We study two other cases: G(A,B)=A1/2(A-1/2BA-1/2)1/2A1/2,\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {G}(A,B)=A^{1/2}(A^{-1/2}BA^{-1/2})^{1/2}A^{1/2},$$\end{document} the Pusz–Woronowicz geometric mean, and G(A,B)=exp(logA+logB2),\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {G}(A,B)=\exp \big (\frac{\log A+\log B}{2}\big ),$$\end{document} the log Euclidean mean. With these choices, d(A, B) is no longer a metric, but it turns out that d2(A,B)\documentclass[12pt]{minimal}
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\begin{document}$$d^2(A,B)$$\end{document} 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.
