Obstructions to extension of Wasserstein distances for variable masses

Lombardini, Luca; Rossi, Francesco

doi:10.1090/proc/16030

Obstructions to extension of Wasserstein distances for variable masses
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by Luca Lombardini and Francesco Rossi PDF

Proc. Amer. Math. Soc. 150 (2022), 4879-4890 Request permission

Abstract:

We study the possibility of defining a distance on the whole space of measures, with the property that the distance between two measures having the same mass is the Wasserstein distance, up to a scaling factor. We prove that, under very weak and natural conditions, if the base space is unbounded, then the scaling factor must be constant, independently of the mass. Moreover, no such distance can exist, if we include the zero measure. Instead, we provide examples with non-constant scaling factors for the case of bounded base spaces.

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