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On uniqueness of an optimal solution to the Kantorovich problem with density constraints

Svetlana Popova

Abstract

We study optimal transportation problems with constraints on densities of transport plans. We obtain a sharp condition for the uniqueness of an optimal solution to the Kantorovich problem with density constraints, namely that the Borel measurable cost function $h(x, y)$ satisfies the following non-degeneracy condition: $h(x, y)$ can not be expressed as a sum of functions $u(x) + v(y)$ on a set of positive measure.

On uniqueness of an optimal solution to the Kantorovich problem with density constraints

Abstract

We study optimal transportation problems with constraints on densities of transport plans. We obtain a sharp condition for the uniqueness of an optimal solution to the Kantorovich problem with density constraints, namely that the Borel measurable cost function satisfies the following non-degeneracy condition: can not be expressed as a sum of functions on a set of positive measure.
Paper Structure (3 sections, 8 theorems, 107 equations)

This paper contains 3 sections, 8 theorems, 107 equations.

Table of Contents

  1. Introduction
  2. Uniqueness of an optimal solution to the Kantorovich problem with density constraints
  3. Non-degeneracy of a cost function with respect to a measure

Key Result

Theorem 2.1

(Existence) Let $(X, \mathcal{A}, \mu)$ and $(Y, \mathcal{B}, \nu)$ be probability spaces. Let $\eta \in \mathcal{P}(X \times Y)$ and $\Phi \in L^1(\eta)$. Suppose that $\Pi_{\Phi}(\mu, \nu; \eta) \neq \varnothing$. Let $h \colon X \times Y \to \mathbb R$ be a bounded from below $\mathcal{A} \otime

Theorems & Definitions (18)

  • Theorem 2.1: BDM21
  • Theorem 2.2
  • proof
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • proof
  • Example 3.3
  • Proposition 3.4
  • Corollary 3.5
  • ...and 8 more