Geometry and inequalities of geometric mean
We study some geometric properties associated with the t-geometric means A ♯ t B:= A 1/2(A −1/2 BA −1/2) t A 1/2 of two n × n positive definite matrices A and B. Some geodesical convexity results with respect to the Riemannian structure of the n × 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 of positive definite matrices is posted.
Keywordsgeometric mean positive definite matrix log majorization geodesics geodesically convex geodesic convex hull unitarily invariant norm
MSC 201015A45 15B48
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