Geodesic
Monotonity of the mean distance for empirical Gaussian samples
Zapiski Nauchnykh Seminarov POMI, Problems of the theory of probability distributions. Part X, Tome 158 (1987), pp. 167-168

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The monotonicity is proved of the mean distance in the sense of Kantorovich for two repeatedly independently obtained dependent Gaussian samples with respect to the natural order in the space of Gaussian centered measures in a finite-dimensional coordinate space. 
@article{ZNSL_1987_158_a17,
     author = {V. N. Sudakov},
     title = {Monotonity of the mean distance for empirical {Gaussian} samples},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {167--168},
     publisher = {mathdoc},
     volume = {158},
     year = {1987},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_1987_158_a17/}
}
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V. N. Sudakov. Monotonity of the mean distance for empirical Gaussian samples. Zapiski Nauchnykh Seminarov POMI, Problems of the theory of probability distributions. Part X, Tome 158 (1987), pp. 167-168. http://geodesic.mathdoc.fr/item/ZNSL_1987_158_a17/