### An extension of a result of Lewis.

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We extend a monotonicity result of Wang and Gong on the product of positive definite matrices in the context of semisimple Lie groups. A similar result on singular values is also obtained.

We give extensions of inequalities of Araki-Lieb-Thirring, Audenaert, and Simon, in the context of semisimple Lie groups.

A generalization of a result of Berezin and Gel’fand in the context of Eaton triples is given. The generalization and its proof are Lie-theoretic free and requires some basic knowledge of nonsmooth analysis. The result is then applied to determine the distance between a point and a G-orbit or its convex hull.We also discuss the derivatives of some orbital functions.

We study some geometric properties associated with the $t$-geometric means $A{\u266f}_{t}B:={A}^{1/2}{\left({A}^{-1/2}B{A}^{-1/2}\right)}^{t}{A}^{1/2}$ of two $n\times n$ positive definite matrices $A$ and $B$. Some geodesical convexity results with respect to the Riemannian structure of the $n\times n$ positive definite matrices are obtained. Several norm inequalities with geometric mean are obtained. In particular, we generalize a recent result of Audenaert (2015). Numerical counterexamples are given for some inequality questions. A conjecture on the geometric mean inequality regarding $m$ pairs...

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