An extremal problem for empirical measures under dependent Gaussian observations

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 $\Sigma=1$.