Geodesic
A counterexample to the conjecture on monotonicity of an integral with respect to Gaussian measure
Zapiski Nauchnykh Seminarov POMI, Probability and statistics. Part 3, Tome 260 (1999), pp. 250-257

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It is shown that for the Kantorovich metrics $\varkappa$ on probability measures for centered Gaussian measures $\gamma$ defined on Euclidean space $E$ of random variables $X$ the integral $$ I(\gamma)=\iint\limits_{E\oplus E}\varkappa(\mathscr L(X_1),\mathscr L(X_2))(\gamma\otimes\gamma)\,d(X_1,X_2), $$ is not always monotonic in $\gamma$. 
@article{ZNSL_1999_260_a16,
     author = {A. V. Sudakov},
     title = {A counterexample to the conjecture on monotonicity of an integral with respect to {Gaussian} measure},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {250--257},
     publisher = {mathdoc},
     volume = {260},
     year = {1999},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_1999_260_a16/}
}
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A. V. Sudakov. A counterexample to the conjecture on monotonicity of an integral with respect to Gaussian measure. Zapiski Nauchnykh Seminarov POMI, Probability and statistics. Part 3, Tome 260 (1999), pp. 250-257. http://geodesic.mathdoc.fr/item/ZNSL_1999_260_a16/