Monotonicity of the matrix geometric mean

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Generalized locally Toeplitz sequences : Theory and applications (2017, background): “It was proved with some effort [13, 62, 72] that the Karcher mean satisfies all the ALM axioms and some further properties, and thus now everyone agrees that the Karcher mean has the right to be called the geometric mean of matrices.”
On the Bures–Wasserstein distance between positive definite matrices (2017, unknown): “See [7] [8] [9] [10] [19] [23] [24] [25] [26].”
Weighted deterministic walks for the least squares mean on Hadamard spaces (2014, background): “The proof was simplified by Bhatia and Karandikar [5] and at the same time Bhatia [3] considers an obvious deterministic sequence generated by the unweighted inductive means and raises the question, whether this deterministic sequence converges to the least squares mean.”