R. E. Barlow, D. J. Bartholomew, J. M. Bremner, and H. D. Brunk, Statistical inference under order restrictions . The theory and application of isotonic regression, 1972.

A. Belloni and R. L. Winkler, On multivariate quantiles under partial orders, The Annals of Statistics, vol.39, issue.2, pp.1125-1179, 2011.
DOI : 10.1214/10-AOS863
URL : http://arxiv.org/abs/0912.5489

Y. Brenier, Polar factorization and monotone rearrangement of vector-valued functions, Communications on Pure and Applied Mathematics, vol.117, issue.4, pp.375-417, 1991.
DOI : 10.1002/cpa.3160440402

G. Carlier, A. Galichon, and F. Santambrogio, From Knothe's Transport to Brenier's Map and a Continuation Method for Optimal Transport, SIAM Journal on Mathematical Analysis, vol.41, issue.6, pp.2554-2576, 2010.
DOI : 10.1137/080740647

P. Chaudhuri, On a Geometric Notion of Quantiles for Multivariate Data, Journal of the American Statistical Association, vol.24, issue.434, pp.862-872, 1996.
DOI : 10.1080/01621459.1996.10476954

V. Chernozhukov, I. Fernandez-val, and A. Galichon, Quantile and Probability Curves without Crossing, Econometrica, vol.78, issue.3, pp.1093-1125, 2010.
URL : https://hal.archives-ouvertes.fr/hal-01052958

F. Cunha, J. Heckman, and S. Schennach, Estimating the technology of cognitive and noncognitive skill formation, Econometrica, vol.78, issue.3, pp.883-931, 2010.
DOI : 10.3386/w15664

H. Dette, N. Neumeyer, and K. Pilz, A simple nonparametric estimator of a strictly monotone regression function, Bernoulli, vol.12, issue.3, pp.469-490, 2006.
DOI : 10.3150/bj/1151525131

K. Doksum, Empirical Probability Plots and Statistical Inference for Nonlinear Models in the Two-Sample Case, The Annals of Statistics, vol.2, issue.2, pp.267-277, 1974.
DOI : 10.1214/aos/1176342662

E. Ducpetiaux, BudgetséconomiquesBudgetséconomiques des classesouvrì eres en Belgique, 1855.

R. M. Dudley and W. Philipp, Invariance principles for sums of Banach space valued random elements and empirical processes, Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete, pp.509-552, 1983.

I. Ekeland, A. Galichon, and M. Henry, COMONOTONIC MEASURES OF MULTIVARIATE RISKS, Mathematical Finance, vol.51, issue.4, pp.109-132, 2012.
DOI : 10.1111/j.1467-9965.2010.00453.x
URL : https://hal.archives-ouvertes.fr/hal-01053550

E. Engel, Die Produktions und Konsumptionsverhältnisse des Königreichs Sachsen, Zeitschrift des Statistischen Bureaus des Königlich Sächsischen Misisteriums des Innerm, pp.1-54, 1857.

M. Hallin, D. Paindaveine, and M. Siman, Multivariate quantiles and multiple-output regression quantiles: From L 1 optimization to halfspace depth, The Annals of Statistics, vol.38, issue.2, pp.635-669, 2010.
DOI : 10.1214/09-AOS723

X. He, Quantile Curves Without Crossing, American Statistician, vol.51, pp.186-192, 1997.
DOI : 10.1080/00031305.1997.10473959

R. Koenker, Quantile Regression Econometric Society Monograph Series 38, 2005.

R. Koenker, quantreg: Quantile Regression. R package version 4.10, 2007.

R. Koenker and G. Bassett, Regression Quantiles, Econometrica, vol.46, issue.1, pp.33-50, 1978.
DOI : 10.2307/1913643

R. Koenker and G. Bassett, Robust Tests for Heteroscedasticity Based on Regression Quantiles, Econometrica, vol.50, issue.1, pp.43-61, 1982.
DOI : 10.2307/1912528

R. Koenker and P. Ng, Inequality constrained quantile regression, pp.418-440, 2005.

V. Koltchinskii, M -estimation, convexity and quantiles, The Annals of Statistics, vol.25, issue.2, pp.435-477, 1997.
DOI : 10.1214/aos/1031833659

L. Kong and I. Mizera, Quantile tomography: using quantiles with multivariate data, Statistica Sinica, vol.22, pp.1589-1610, 2012.
DOI : 10.5705/ss.2010.224

R. J. Mccann, Existence and uniqueness of monotone measure-preserving maps, Duke Mathematical Journal, vol.80, issue.2, pp.309-324, 1995.
DOI : 10.1215/S0012-7094-95-08013-2

B. Laine, Depth contours as multivariate quantiles: a directional approach, 2001.

E. Lehmann, Nonparametrics: statistical methods based on ranks, 1974.

L. Play and F. , Les Ouvriers Européens. Etudes sur les travaux, la vie domestique et la condition morale des populationsouvrì eres de l'Europe, 1855.

R. Matzkin, Nonparametric Estimation of Nonadditive Random Functions, Econometrica, vol.71, issue.5, pp.1339-1375, 2003.
DOI : 10.1111/1468-0262.00452

R. Development and C. Team, R: A language and environment for statistical computing. R Foundation for Statistical Computing, 2007.

J. V. Ryff, Measure preserving transformations and rearrangements, Journal of Mathematical Analysis and Applications, vol.31, issue.2, pp.449-458, 1970.
DOI : 10.1016/0022-247X(70)90038-7
URL : http://doi.org/10.1016/0022-247x(70)90038-7

R. Serfling, Nonparametric multivariate descriptive measures based on spatial quantiles, Journal of Statistical Planning and Inference, vol.123, issue.2, pp.259-278, 2004.
DOI : 10.1016/S0378-3758(03)00156-3

J. Tukey, Mathematics and the picturing of data, Proc. 1975 International Congress of Mathematicians, pp.523-531, 1975.

C. Villani, Topics in Optimal Transportation, 2003.
DOI : 10.1090/gsm/058

C. Villani, Optimal transport: Old and New. Grundlehren der mathematischen Wissenschaften, 2009.
DOI : 10.1007/978-3-540-71050-9

Y. Wei, An Approach to Multivariate Covariate-Dependent Quantile Contours With Application to Bivariate Conditional Growth Charts, Journal of the American Statistical Association, vol.103, issue.481, pp.397-409, 2008.
DOI : 10.1198/016214507000001472

K. Yu and M. C. Jones, Local Linear Quantile Regression, Journal of the American Statistical Association, vol.25, issue.441, pp.228-237, 1998.
DOI : 10.1080/01621459.1998.10474104

C. and U. Cnrs, 75775 Paris Cedex 16, FRANCE E-mail address: carlier@ceremade.dauphine.fr Department of Economics, MIT, 50 Memorial Drive, E-mail address: vchern@mit.edu Sciences Po, pp.52-361