Existence and uniqueness of optimal transport maps

Existence and Uniqueness of Optimal Transport Maps

Let (X, d,m) be a proper, non-branching, metric measure space. We show existence and uniqueness of optimal transport maps for cost written as non-decreasing and strictly convex functions of the distance, provided (X, d,m) satisfies a new weak property concerning the behavior of m under the shrinking of sets to points, see Assumption 1. This in particular covers spaces satisfying the measure con...

Existence, Uniqueness, and Regularity of Optimal Transport Maps

Adapting some techniques and ideas of McCann [8], we extend a recent result with Fathi [6] to yield existence and uniqueness of a unique transport map in very general situations, without any integrability assumption on the cost function. In particular this result applies for the optimal transportation problem on a n-dimensional non-compact manifold M with a cost function induced by a C2-Lagrang...

Existence and uniqueness of optimal maps on Alexandrov spaces

The purpose of this paper is to show that in a finite dimensional metric space with Alexandrov’s curvature bounded below, Monge’s transport problem for the quadratic cost admits a unique solution.

Existence and Uniqueness of Optimal Matrix Scalings

The problem of finding a diagonal similarity scaling to minimize the scaled singular value of a matrix arises frequently in robustness analysis of control systems. It is shown here that the set of optimal diagonal scalings is nonempty and bounded if and only if the matrix that is being scaled is irreducible. For an irreducible matrix, a sufficient condition is derived for the uniqueness of the ...

Existence and Uniqueness of Monotone Measure - Preserving Maps