**Memoirs of the American Mathematical Society**

2018;
112 pp;
Softcover

MSC: Primary 28; 49;

**Print ISBN: 978-1-4704-2766-5
Product Code: MEMO/251/1197**

List Price: $78.00

AMS Member Price: $47.00

MAA Member Price: $70.20

**Electronic ISBN: 978-1-4704-4278-1
Product Code: MEMO/251/1197.E**

List Price: $78.00

AMS Member Price: $47.00

MAA Member Price: $70.20

# On Sudakov’s Type Decomposition of Transference Plans with Norm Costs

Share this page
*Stefano Bianchini; Sara Daneri*

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.

#### Table of Contents

# Table of Contents

## On Sudakov's Type Decomposition of Transference Plans with Norm Costs

- Cover Cover11
- Title page i2
- Chapter 1. Introduction 18
- Chapter 2. General notations and definitions 2128
- 1. Functions and multifunctions 2128
- 2. Affine subspaces, convex sets and norms 2229
- 3. Measures and disintegration 2431
- 4. Optimal transportation problems 2633
- 5. Linear preorders, uniqueness and optimality 2734
- 6. Optimal transportation problems with convex norm and cone costs 3340
- 7. Transportation problems with convex norms and cone costs on Lipschitz graphs 3441
- 8. Optimal transportation problems on directed locally affine partitions 3542
- 9. From directed partitions to directed fibrations 4148

- Chapter 3. Directed locally affine partitionson cone-Lipschitz foliations 4956
- 1. Convex cone-Lipschitz graphs 4956
- 2. Convex cone-Lipschitz foliations 5158
- 3. Regular transport sets and residual set 5461
- 4. Super/subdifferential directed partitions of regular sets 6067
- 5. Analysis of the residual set 6370
- 6. Optimal transportation on c-Lipschitz foliations 6572
- 7. Dimensional reduction on directed partitions via cone approximation property 6976
- 8. Model sets of directed segments 6976
- 9. k-dimensional model sets 7481
- 10. k-dimensional sheaf sets and D-cylinders 7683
- 11. Negligibility of initial/final points 7784

- Chapter 4. Proof of Theorem 1.1 8188
- Chapter 5. From 𝐂^{𝐤}-fibrations to linearly ordered 𝐂^{𝐤}-Lipschitz foliations 8592
- Chapter 6. Proof of Theorems 1.2-1.6. 101108
- Appendix A. Minimality of equivalence relations 103110
- B. Notation 105112
- C. Index of definitions 109116
- Bibliography 111118
- Back Cover Back Cover1124