The classical Kantorovich-Rubinstein duality guarantees coincidence between metrics on the space of probability distributions defined on the one hand via transport plans (couplings) and on the other hand via price functions. Both constructions have been lifted to the level of generality of set functors, with the coupling-based construction referred to as the Wasserstein lifting, and the price-function-based construction as the Kantorovich lifting, both based on a choice of quantitative modalities for the given functor. It is known that every Wasserstein lifting can be expressed as a Kantorovich lifting; however, the latter in general needs to use additional modalities. We give an example showing that this cannot be avoided in general. We refer to cases in which the same modalities can be used as satisfying the generalized Kantorovich-Rubinstein duality. We establish the generalized Kantorovich-Rubinstein duality in this sense for two important cases: The L\'evy-Prokhorov distance on distributions, which finds wide-spread applications in machine learning due to its favourable stability properties, and the standard metric on convex sets of distributions that arises by combining the Hausdorff and Wasserstein distances.

翻译：经典的Kantorovich-Rubinstein对偶性保证了概率分布空间上两种度量定义的一致性：一种通过传输方案（耦合）定义，另一种通过价格函数定义。这两种构造已被提升到集合函子的一般性层面，其中基于耦合的构造称为Wasserstein提升，而基于价格函数的构造称为Kantorovich提升，两者均基于对给定函子的量化模态的选择。已知每个Wasserstein提升均可表示为Kantorovich提升；然而，后者通常需要使用额外的模态。我们给出一个示例，表明在一般情况下这无法避免。我们将能够使用相同模态的情形称为满足广义Kantorovich-Rubinstein对偶性。我们在这一意义上为两个重要案例建立了广义Kantorovich-Rubinstein对偶性：一是分布上的Lévy-Prokhorov距离，由于其良好的稳定性在机器学习中得到广泛应用；二是通过结合Hausdorff距离与Wasserstein距离产生的分布凸集上的标准度量。