
On the Wasserstein distance between mutually singular measures
Buttazzo, Giuseppe; Carlier, Guillaume; Laborde, Maxime (2018), On the Wasserstein distance between mutually singular measures, Advances in Calculus of Variations, 13, 2, p. 141–154. 10.1515/acv-2017-0036
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Article accepté pour publication ou publiéDate
2018Journal name
Advances in Calculus of VariationsVolume
13Number
2Publisher
De Gruyter
Pages
141–154
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Show full item recordAuthor(s)
Buttazzo, GiuseppeDipartimento di Matematica [Pisa]
Carlier, Guillaume
CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
Laborde, Maxime
CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
Abstract (EN)
We study the Wasserstein distance between two measures µ, ν which are mutually singular. In particular, we are interested in minimization problems of the form W (µ, A) = inf W (µ, ν) : ν ∈ A where µ is a given probability and A is contained in the class µ ⊥ of probabilities that are singular with respect to µ. Several cases for A are considered; in particular, when A consists of L 1 densities bounded by a constant, the optimal solution is given by the characteristic function of a domain. Some regularity properties of these optimal domains are also studied. Some numerical simulations are included, as well as the double minimization problem min P (B) + kW (A, B) : |A ∩ B| = 0, |A| = |B| = 1 , where k > 0 is a fixed constant, P (A) is the perimeter of A, and both sets A, B may vary.Subjects / Keywords
Wasserstein distance; singular measures; perimeter penalizationRelated items
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