Journal of Lie Theory

no. 3
A Geometric Mean for Symmetric Spaces of Noncompact Type
Journal of Lie Theory, Volume 24 (2014) no. 3, pp. 725-736
The concept of the t-geometric mean of two positive definite matrices is extended to symmetric spaces of noncompact type. The t-geometric mean of two points in such a symmetric space yields the unique geodesic joining the points and the geometric mean is the midpoint. A parametrization of the geodesic in terms of the two points is given. Inequalities about geometric mean and geodesic triangle are given in terms of Kostant's pre-order on semisimple Lie groups as well as on their Lie algebras.
DOI: 10.5802/jolt.804
Classification: 15A45, 15A48, 53C35
Keywords: Geometric mean, positive definite matrices, symmetric spaces, semisimple Lie groups, geodesics, log majorization, Kostant's order 
@article{JOLT_2014_24_3_a6,
     author = {M. Liao and X. Liu and T.-Y. Tam},
     title = {A {Geometric} {Mean} for {Symmetric} {Spaces} of {Noncompact} {Type}},
     journal = {Journal of Lie Theory},
     pages = {725--736},
     year = {2014},
     volume = {24},
     number = {3},
     doi = {10.5802/jolt.804},
     zbl = {1331.15011},
     url = {https://jolt.centre-mersenne.org/articles/10.5802/jolt.804/}
}
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M. Liao; X. Liu; T.-Y. Tam. A Geometric Mean for Symmetric Spaces of Noncompact Type. Journal of Lie Theory, Volume 24 (2014) no. 3, pp. 725-736. doi: 10.5802/jolt.804

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