Geodesic
Wasserstein and weighted metrics for multidimensional Gaussian distributions
Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 23 (2023) no. 4, pp. 422-434.

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We present a number of low and upper bounds for Lévy – Prokhorov, Wasserstein, Frechét, and Hellinger distances between probability distributions of the same or different dimensions. The weighted (or context-sensitive) total variance and Hellinger distances are introduced. The upper and low bounds for these weighted metrics are proved. The low bounds for the minimum of different errors in sensitive hypothesis testing are proved. 
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M. Y. Kelbert; Y. Suhov. Wasserstein and weighted metrics for multidimensional Gaussian distributions. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 23 (2023) no. 4, pp. 422-434. http://geodesic.mathdoc.fr/item/ISU_2023_23_4_a0/

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