Publications and Preprints

An attractive candidate for the geometric mean of $m$ positive definite matrices $A_1,\ldots , A_m$ is their Riemannian barycentre $G$. One of its important properties, monotonicity in the $m$ arguments, has been established recently by J. Lawson and Y. Lim. We give a much simpler proof of this result, and prove some other inequalities. One of these says that, for every unitarily invariant norm, $||| G |||$ is not bigger than the geometric mean of $|||A_1|||, \ldots, |||A_m|||$.