Geodesic
An extremal problem for empirical measures under dependent Gaussian observations
Zapiski Nauchnykh Seminarov POMI, Problems of the theory of probability distributions. Part IX, Tome 142 (1985), pp. 164-166

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One describes a class of metrics $\rho$ in the space of probability distributions on the line, for which the minimum of the mean value of the random variablep $\rho(F_X^*, F_Y^*)$, where $X$, $Y$ are independent random variables, distributed according to the Gauss law $N(0,\Sigma)$, $\Sigma\le1$, is attained at $\Sigma=1$. 
@article{ZNSL_1985_142_a18,
     author = {V. N. Sudakov},
     title = {An extremal problem for empirical measures under dependent {Gaussian} observations},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {164--166},
     publisher = {mathdoc},
     volume = {142},
     year = {1985},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_1985_142_a18/}
}
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V. N. Sudakov. An extremal problem for empirical measures under dependent Gaussian observations. Zapiski Nauchnykh Seminarov POMI, Problems of the theory of probability distributions. Part IX, Tome 142 (1985), pp. 164-166. http://geodesic.mathdoc.fr/item/ZNSL_1985_142_a18/