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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

### 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.

- Luigi Ambrosio,
*Lecture notes on optimal transport problems*, Mathematical aspects of evolving interfaces (Funchal, 2000) Lecture Notes in Math., vol. 1812, Springer, Berlin, 2003, pp. 1–52. MR**2011032**, DOI 10.1007/978-3-540-39189-0_{1} - L. Ambrosio and N. Gigli. A user\rqs guide to optimal transport. Technical report.
- L. Ambrosio, B. Kirchheim, and A. Pratelli,
*Existence of optimal transport maps for crystalline norms*, Duke Math. J.**125**(2004), no. 2, 207–241. MR**2096672**, DOI 10.1215/S0012-7094-04-12521-7 - Luigi Ambrosio and Aldo Pratelli,
*Existence and stability results in the $L^1$ theory of optimal transportation*, Optimal transportation and applications (Martina Franca, 2001) Lecture Notes in Math., vol. 1813, Springer, Berlin, 2003, pp. 123–160. MR**2006307**, DOI 10.1007/978-3-540-44857-0_{5} - Mathias Beiglböck, Martin Goldstern, Gabriel Maresch, and Walter Schachermayer,
*Optimal and better transport plans*, J. Funct. Anal.**256**(2009), no. 6, 1907–1927. MR**2498564**, DOI 10.1016/j.jfa.2009.01.013 - Stefano Bianchini and Laura Caravenna,
*On the extremality, uniqueness and optimality of transference plans*, Bull. Inst. Math. Acad. Sin. (N.S.)**4**(2009), no. 4, 353–454. MR**2582736** - Stefano Bianchini and Matteo Gloyer,
*On the Euler-Lagrange equation for a variational problem: the general case II*, Math. Z.**265**(2010), no. 4, 889–923. MR**2652541**, DOI 10.1007/s00209-009-0547-2 - Stefano Bianchini and Matteo Gloyer,
*An estimate on the flow generated by monotone operators*, Comm. Partial Differential Equations**36**(2011), no. 5, 777–796. MR**2769108**, DOI 10.1080/03605302.2010.534224 - Yann Brenier,
*Polar factorization and monotone rearrangement of vector-valued functions*, Comm. Pure Appl. Math.**44**(1991), no. 4, 375–417. MR**1100809**, DOI 10.1002/cpa.3160440402 - Luis A. Caffarelli, Mikhail Feldman, and Robert J. McCann,
*Constructing optimal maps for Monge’s transport problem as a limit of strictly convex costs*, J. Amer. Math. Soc.**15**(2002), no. 1, 1–26. MR**1862796**, DOI 10.1090/S0894-0347-01-00376-9 - Laura Caravenna,
*A proof of Monge problem in $\Bbb R^n$ by stability*, Rend. Istit. Mat. Univ. Trieste**43**(2011), 31–51. MR**2933122** - Laura Caravenna,
*A proof of Sudakov theorem with strictly convex norms*, Math. Z.**268**(2011), no. 1-2, 371–407. MR**2805441**, DOI 10.1007/s00209-010-0677-6 - L. Caravenna. An existence result for the Monge problem in $\R ^n$ with norm cost function. Preprint SISSA, 2009. http://cvgmt.sns.it/media/doc/paper/1303/sel.partMonge.pdf
- Guillaume Carlier, Luigi De Pascale, and Filippo Santambrogio,
*A strategy for non-strictly convex transport costs and the example of $\|X-Y\|^P$ in $\Bbb R^2$*, Commun. Math. Sci.**8**(2010), no. 4, 931–941. MR**2744914** - L. Caravenna and S. Daneri,
*The disintegration of the Lebesgue measure on the faces of a convex function*, J. Funct. Anal.**258**(2010), no. 11, 3604–3661. MR**2606867**, DOI 10.1016/j.jfa.2010.01.024 - Arrigo Cellina and Stefania Perrotta,
*On the validity of the maximum principle and of the Euler-Lagrange equation for a minimum problem depending on the gradient*, SIAM J. Control Optim.**36**(1998), no. 6, 1987–1998. MR**1638928**, DOI 10.1137/S0363012996311319 - Thierry Champion and Luigi De Pascale,
*The Monge problem for strictly convex norms in $\Bbb R^d$*, J. Eur. Math. Soc. (JEMS)**12**(2010), no. 6, 1355–1369. MR**2734345**, DOI 10.4171/JEMS/234 - Thierry Champion and Luigi De Pascale,
*The Monge problem in $\Bbb R^d$*, Duke Math. J.**157**(2011), no. 3, 551–572. MR**2785830**, DOI 10.1215/00127094-1272939 - S. Daneri.
*Dimensional Reduction and Approximation of Measures and Weakly Differentiable Homeomorphisms*. PhD thesis, SISSA, 2011. - L.C. Evans and W. Gangbo. Differential equations methods for the monge-kantorovich mass transfer problem.
*Current Developments in Mathematics*, pages 65–126, 1997. - D. H. Fremlin.
*Measure Theory*, volume 4. Torres Fremlin, 2002. 332Tb. - D. H. Fremlin,
*Measure theory. Vol. 1*, Torres Fremlin, Colchester, 2004. The irreducible minimum; Corrected third printing of the 2000 original. MR**2462519** - C. Jimenez and F. Santambrogio,
*Optimal transportation for a quadratic cost with convex constraints and applications*, J. Math. Pures Appl. (9)**98**(2012), no. 1, 103–113 (English, with English and French summaries). MR**2935372**, DOI 10.1016/j.matpur.2012.01.002 - Hans G. Kellerer,
*Duality theorems for marginal problems*, Z. Wahrsch. Verw. Gebiete**67**(1984), no. 4, 399–432. MR**761565**, DOI 10.1007/BF00532047 - Svetlozar T. Rachev and Ludger Rüschendorf,
*Mass transportation problems. Vol. II*, Probability and its Applications (New York), Springer-Verlag, New York, 1998. Applications. MR**1619171** - A. M. Srivastava.
*A course on Borel sets*. Springer, 1998. - V. N. Sudakov,
*Geometric problems in the theory of infinite-dimensional probability distributions*, Proc. Steklov Inst. Math.**2**(1979), i–v, 1–178. Cover to cover translation of Trudy Mat. Inst. Steklov 141 (1976). MR**530375** - Neil S. Trudinger and Xu-Jia Wang,
*On the Monge mass transfer problem*, Calc. Var. Partial Differential Equations**13**(2001), no. 1, 19–31. MR**1854255**, DOI 10.1007/PL00009922 - Cédric Villani,
*Optimal transport*, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 338, Springer-Verlag, Berlin, 2009. Old and new. MR**2459454**