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The Monge-Kantorovich mass-transportation problem has been shown to be fundamental for various basic problems in analysis and geometry in recent years. Shen and Zheng (2010) proposed a probability method to transform the celebrated Monge-Kantorovich problem in a bounded region of the Euclidean plane into a Dirichlet boundary problem associated to a nonlinear elliptic equation. Their results are original and sound, however, their arguments leading to the main results are skipped and difficult to follow. In the present paper, we adopt a different approach and give a short and easy-followed detailed proof for their main results.
We study the entropic regularization of the optimal transport problem in dimension 1 when the cost function is the distance c(x, y) = |y -- x|. The selected plan at the limit is, among those which are optimal for the non-penalized problem, the most d
In this paper we study the local linearization of the Hellinger--Kantorovich distance via its Riemannian structure. We give explicit expressions for the logarithmic and exponential map and identify a suitable notion of a Riemannian inner product. Sam
We study the barycenter of the Hellinger--Kantorovich metric over non-negative measures on compact, convex subsets of $mathbb{R}^d$. The article establishes existence, uniqueness (under suitable assumptions) and equivalence between a coupled-two-marg
In this note, we study the nonexpansive properties based on arbitrary variable metric and explore the connections between firm nonexpansiveness, cocoerciveness and averagedness. A convergence rate analysis for the associated fixed-point iterations is
We present some partial results regarding subadditivity of maximal shifts in finite graded free resolutions.