Geodesic
Hausdorff metric structure in the space of probability measures
Zapiski Nauchnykh Seminarov POMI, Studies in mathematical statistics. Part 3, Tome 87 (1979), pp. 87-103

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The Hausdorrf distance between two bounded functions on the real line is defined by B. Sendov and B. Penkov. It is knows 2 that the Levy distance between distribution functions and the Hausdorff distance coinside. The connection of the structure of Levy–Prokhorov's distance with that of the Hausdorff distance is analysed in this paper. 
@article{ZNSL_1979_87_a8,
     author = {S. T. Rachev},
     title = {Hausdorff metric structure in the space of probability measures},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {87--103},
     publisher = {mathdoc},
     volume = {87},
     year = {1979},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_1979_87_a8/}
}
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S. T. Rachev. Hausdorff metric structure in the space of probability measures. Zapiski Nauchnykh Seminarov POMI, Studies in mathematical statistics. Part 3, Tome 87 (1979), pp. 87-103. http://geodesic.mathdoc.fr/item/ZNSL_1979_87_a8/