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Geometry and inequalities of geometric mean

Authors: 

Trung Hoa Dinh*, Sima Ahsani, Tin-Yau Tam 

Source title: 
Czechoslovak Mathematical Journal, 66(3): 777-792, 2016 (ISI)
Academic year of acceptance: 
2016-2017
Abstract: 

We study some geometric properties associated with the t-geometric means At 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.