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On Sudakov’s type decomposition of transference plans with norm costs

About this Title

Stefano Bianchini, SISSA, via Bonomea 265, IT-34136 Trieste (ITALY) and Sara Daneri, FAU Erlangen, Department Mathematik, Cauerstr. 11, D-91058 Erlangen (GERMANY)

Publication: Memoirs of the American Mathematical Society
Publication Year: 2018; Volume 251, Number 1197
ISBNs: 978-1-4704-2766-5 (print); 978-1-4704-4278-1 (online)
DOI: https://doi.org/10.1090/memo/1197
Published electronically: November 6, 2017
MSC: Primary 28A50, 49Q20

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Table of Contents

Chapters

• 1. Introduction
• 2. General notations and definitions
• 3. Directed locally affine partitionson cone-Lipschitz foliations
• 4. Proof of Theorem 1.1
• 5. From ${\tilde {\mathbf C}}^k$-fibrations to linearly ordered ${\tilde {\mathbf C}}^k$-Lipschitz foliations
• 6. Proof of Theorems -.
• A. Minimality of equivalence relations
• B. Notation
• C. Index of definitions

Abstract

We consider the original strategy proposed by Sudakov for solving the Monge transportation problem with norm cost $|\cdot |_{D^*}$ \[ min\bigg{ ∫|\mathtt T(x) - x|_{D^{*}} d𝜇(x), \mathtt T : \mathbb{R}^{d} →\mathbb{R}^{d}, 𝜈= \mathtt T_{#} 𝜇\bigg},\] 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 $\mathtt T_\alpha : Z_\alpha \rightarrow \mathbb {R}^d$ 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 we 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.

The analysis requires

(1)$\,$ the study of the transportation problem on directed locally affine partitions $\{Z^k_\alpha ,C^k_\alpha \}_{k,\alpha }$ of $\mathbb {R}^d$, i.e. sets $Z^k_{\alpha } \subset \mathbb {R}^d$ which are relatively open in their $k$-dimensional affine hull and on which the transport occurs only along directions belonging to a cone $C^k_\alpha$;

(2)$\,$ the proof of the absolute continuity w.r.t. the suitable $k$-dimensional Hausdorff measure of the disintegration of $\mathcal {L}^d$ on these directed locally affine partitions;

(3)$\,$ the definition of cyclically connected sets w.r.t. a family of transportation plans with finite cone costs;

(4)$\,$ the proof of the existence of cyclically connected directed locally affine partitions for transport problems with cost functions which are indicator functions of cones and no potentials can be constructed.