Matrix versions of the Hellinger distance

On the space of positive definite matrices, we consider distance functions of the form d(A, B) = [tr A(A, B) - tr G(A, B)] ^{1 / 2}, where A(A, B) is the arithmetic mean and G(A, B) is one of the different versions of the geometric mean. When G(A, B) = A^{1 / 2}B^{1 / 2} this distance is ‖ A^{1 / 2}- B^{1 / 2}‖ ₂, and when G(A,B)=(A1/2BA1/2)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, the Pusz–Woronowicz geometric mean, and G(A,B)=exp(logA+logB2), the log Euclidean mean. With these choices, d(A, B) is no longer a metric, but it turns out that d²(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.