Geodesic
An estimate of distances between finite dimensional symmetric spaces
Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part IX, Tome 92 (1979), pp. 268-273

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If $E_1$, $E_2$ are two $n$-dimensional symmetric spaces then the Banach–Mazur distance between them satisfies the inequality $d(E_1,E_2)\le cn^{1/2}\log^4n$, where $C$ is an absolute constant. 
@article{ZNSL_1979_92_a18,
     author = {E. D. Gluskin},
     title = {An estimate of distances between finite dimensional symmetric spaces},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {268--273},
     publisher = {mathdoc},
     volume = {92},
     year = {1979},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_1979_92_a18/}
}
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E. D. Gluskin. An estimate of distances between finite dimensional symmetric spaces. Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part IX, Tome 92 (1979), pp. 268-273. http://geodesic.mathdoc.fr/item/ZNSL_1979_92_a18/