Abstract We study the Wasserstein distance between two measures μ , ν {\mu,\nu} which are mutually singular. In particular, we are interested in minimization problems of the form W ⁢ ( μ , 𝒜 ) = inf ⁡ { W ⁢ ( μ , ν ) : ν ∈ 𝒜 } , W(\mu,\mathcal{A})=\inf\{W(\mu,\nu):\nu\in\mathcal{A}\}, where μ is a given probability and 𝒜 {\mathcal{A}} is contained in the class μ ⊥ {\mu^{\perp}} of probabilities that are singular with respect to μ. Several cases for 𝒜 {\mathcal{A}} are considered; in particular, when 𝒜 {\mathcal{A}} consists of L 1 {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 ) + k ⁢ W ⁢ ( A , B ) : | A ∩ B | = 0 , | A | = | B | = 1 } , \min\{P(B)+kW(A,B):|A\cap B|=0,\,|A|=|B|=1\}, where k > 0 {k>0} is a fixed constant, P ⁢ ( A ) {P(A)} is the perimeter of A, and both sets A , B {A,B} may vary.

中文翻译：

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关于相互奇异测度之间的Wasserstein距离

摘要 我们研究了互奇异的两个度量 μ , ν {\mu,\nu} 之间的 Wasserstein 距离。特别是，我们对 W ⁢ ( μ , 𝒜 ) = inf ⁡ { W ⁢ ( μ , ν ) : ν ∈ 𝒜 } , W(\mu,\mathcal{A})=\inf 形式的最小化问题感兴趣\{W(\mu,\nu):\nu\in\mathcal{A}\}，其中 μ 是给定的概率，而 𝒜 {\mathcal{A}} 包含在类 μ ⊥ {\mu^{ \perp}} 的关于 μ 的奇异概率。考虑了 𝒜 {\mathcal{A}} 的几种情况；特别是，当 𝒜 {\mathcal{A}} 由 L 1 {L^{1}} 密度组成，且密度为常数时，最优解由域的特征函数给出。还研究了这些最优域的一些规律性特性。包括一些数值模拟，以及双重最小化问题 min ⁡ { P ⁢ ( B ) + k ⁢ W ⁢ ( A , B ) ：| A∩B | = 0 , | 一个 | = | 乙 | = 1 } , \min\{P(B)+kW(A,B):|A\cap B|=0,\,|A|=|B|=1\}, 其中 k > 0 {k> 0} 是一个固定常数，P ⁢ ( A ) {P(A)} 是 A 的周长，并且两个集合 A , B {A,B} 可能会有所不同。