Structure preserving matrix means on the Marcus–Minc stochastic matrices

In this paper we show that the set PSm of all m × m positive definite stochastic matrices with diagonal entries bounded above by is stable under the weighted geometric mean operation. It is further shown that PSm is stable for some well-known multivariable matrix means; the least squares means for the Riemannian trace metric and the Kullback–Leibler divergence on the convex cone of m × m positive definite matrices, and ALM (Ando–Li–Mathias) and BMP (Bini–Meini–Poloni) geometric means.