Geometrical significance of the Lowner-Heinz inequality

It is proven that the Lowner-Heinz inequality parallel to A(t)B(t)parallel to less than or equal to parallel to AB parallel to(t), valid for all posit ive invertible operators A, B on the Hilbert space H and t is an element of [0, 1], has equivalent forms related to the Finsler structure of the space of positive invertible elements of L(H) or, more generally, of a unital C* -algebra. In particular, the Lowner-Heinz inequality is equivalent to some type of "nonpositive curvature" property of that space.