Geodesic
Geometry and inequalities of geometric mean
Czechoslovak Mathematical Journal, Tome 66 (2016) no. 3, pp. 777-792
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We study some geometric properties associated with the $t$-geometric means $A\sharp _{t}B := A^{1/2}(A^{-1/2}BA^{-1/2})^{t}A^{1/2}$ 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 of positive definite matrices is posted.
We study some geometric properties associated with the $t$-geometric means $A\sharp _{t}B := A^{1/2}(A^{-1/2}BA^{-1/2})^{t}A^{1/2}$ 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 of positive definite matrices is posted.
DOI : 10.1007/s10587-016-0292-8
Classification : 15A45, 15B48
Keywords: geometric mean; positive definite matrix; log majorization; geodesics; geodesically convex; geodesic convex hull; unitarily invariant norm 
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Dinh, Trung Hoa; Ahsani, Sima; Tam, Tin-Yau. Geometry and inequalities of geometric mean. Czechoslovak Mathematical Journal, Tome 66 (2016) no. 3, pp. 777-792. doi: 10.1007/s10587-016-0292-8

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