Geodesic
Two ways to define compatible metrics on the simplex of measures
Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial methods. Part XXII, Tome 411 (2013), pp. 38-48

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We introduce two general methods of lifting a metric on a space to the simplex of probability measures on the metric space. The first one is the method of transportation plans, or the coupling method; the second one is the method of considering norms dual to the restrictions of the Lipschitz norm to subspaces. The intersection of these two classes of metrics consists of the Kantorovich metric. 
@article{ZNSL_2013_411_a1,
     author = {A. M. Vershik},
     title = {Two ways to define compatible metrics on the simplex of measures},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {38--48},
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     volume = {411},
     year = {2013},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2013_411_a1/}
}
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A. M. Vershik. Two ways to define compatible metrics on the simplex of measures. Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial methods. Part XXII, Tome 411 (2013), pp. 38-48. http://geodesic.mathdoc.fr/item/ZNSL_2013_411_a1/