Multi-variable weighted geometric means of positive definite matrices

We define a family of weighted geometric means {G(t;ω;A)}t∈[0,1]n where ω and A vary over all positive probability vectors in Rn and n-tuples of positive definite matrices resp. Each of these weighted geometric means interpolates between the weighted ALM (t=0n) and BMP (t=1n) geometric means (ALM and BMP geometric means have been defined by Ando–Li–Mathias and Bini–Meini–Poloni, respectively.) We show that the weighted geometric means satisfy multidimensional versions of all properties that one would expect for a two-variable weighted geometric mean.