On Sudakov's type decomposition of transference plans with norm costs / Stefano Bianchini, Sara Daneri.

Format:

Book

Author/Creator:

Bianchini, Stefano, 1970- author.

Daneri, Sara, 1983- author.

Series:

Memoirs of the American Mathematical Society ; Volume 251, Number 1197.

Memoirs of the American Mathematical Society ; Volume 251, Number 1197

Language:

English

Subjects (All):

Decomposition (Mathematics).

Mathematical optimization.

Physical Description:

1 online resource (124 pages).

Edition:

1st ed.

Place of Publication:

Providence, Rhode Island : American Mathematical Society, [2018]

Summary:

The authors consider the original strategy proposed by Sudakov for solving the Monge transportation problem with norm cost |\cdot|_{D^*} \min \bigg\{ \int |\mathtt T(x) - x|_{D^*} d\mu(x), \ \mathtt T : \mathbb{R}^d \to \mathbb{R}^d, \ \nu = \mathtt T_\# \mu \bigg\}, with \mu, \nu probability measures in \mathbb{R}^d and \mu absolutely continuous w.r.t. \mathcal{L}^d. The key idea in this approach is to decompose (via disintegration of measures) the Kantorovich optimal transportation problem into a family of transportation problems in Z_\alpha\times \mathbb{R}^d, where \{Z_\alpha\}_{\alpha\in\mathfrak{A}} \subset \mathbb{R}^d are disjoint regions such that the construction of an optimal map \mathtt T_\alpha : Z_\alpha \to \mathbb{R}^d is simpler than in the original problem, and then to obtain \mathtt T by piecing together the maps \mathtt T_\alpha. When the norm |{\cdot}|_{D^*} is strictly convex, the sets Z_\alpha are a family of 1-dimensional segments determined by the Kantorovich potential called optimal rays, while the existence of the map \mathtt T_\alpha is straightforward provided one can show that the disintegration of \mathcal L^d (and thus of \mu) on such segments is absolutely continuous w.r.t. the 1-dimensional Hausdorff measure. When the norm |{\cdot}|_{D^*} is not strictly convex, the main problems in this kind of approach are two: first, to identify a suitable family of regions \{Z_\alpha\}_{\alpha\in\mathfrak{A}} on which the transport problem decomposes into simpler ones, and then to prove the existence of optimal maps. In this paper the authors show how these difficulties can be overcome, and that the original idea of Sudakov can be successfully implemented. The results yield a complete characterization of the Kantorovich optimal transportation problem, whose straightforward corollary is the solution of the Monge problem in

each set Z_\alpha and then in \mathbb{R}^d. The strategy is sufficiently powerful to be applied to other optimal transportation problems.

Contents:

Cover

Title page

Chapter 1. Introduction

1. Sudakov's strategy in the strictly convex case

2. Sudakov's strategy in the general convex case

3. Structure of the paper

Chapter 2. General notations and definitions

1. Functions and multifunctions

2. Affine subspaces, convex sets and norms

3. Measures and disintegration

4. Optimal transportation problems

5. Linear preorders, uniqueness and optimality

6. Optimal transportation problems with convex norm and cone costs

7. Transportation problems with convex norms and cone costs on Lipschitz graphs

8. Optimal transportation problems on directed locally affine partitions

9. From directed partitions to directed fibrations

Chapter 3. Directed locally affine partitionson cone-Lipschitz foliations

1. Convex cone-Lipschitz graphs

2. Convex cone-Lipschitz foliations

3. Regular transport sets and residual set

4. Super/subdifferential directed partitions of regular sets

5. Analysis of the residual set

6. Optimal transportation on c-Lipschitz foliations

7. Dimensional reduction on directed partitions via cone approximation property

8. Model sets of directed segments

9. k-dimensional model sets

10. k-dimensional sheaf sets and D-cylinders

11. Negligibility of initial/final points

Chapter 4. Proof of Theorem 1.1

Chapter 5. From ^{ }-fibrations to linearly ordered ^{ }-Lipschitz foliations

1. Construction of a (c,m,n)-compatible linear preorder

2. Minimal (c,m,n)-compatible linear preorder

3. Cone approximation property for linearly ordered C-Lipschitz foliations

Chapter 6. Proof of Theorems 1.2-1.6.

1. Proof of Theorems 1.6 and 1.2

2. Proof of Theorems 1.3 and 1.5

Appendix A. Minimality of equivalence relations

B. Notation

C. Index of definitions

Bibliography

Back Cover.

Notes:

Includes bibliographical references and index.

Description based on print version record.

ISBN:

1-4704-4278-7