# American Institute of Mathematical Sciences

April  2014, 34(4): 1511-1532. doi: 10.3934/dcds.2014.34.1511

## Hessian metrics, $CD(K,N)$-spaces, and optimal transportation of log-concave measures

 1 Higher School of Economics, Faculty of Mathematics, 117312, Vavilova 7, Moscow, Russian Federation

Received  November 2012 Revised  April 2013 Published  October 2013

We study the optimal transportation mapping $\nabla \Phi : \mathbb{R}^d \mapsto \mathbb{R}^d$ pushing forward a probability measure $\mu = e^{-V} \ dx$ onto another probability measure $\nu = e^{-W} \ dx$. Following a classical approach of E. Calabi we introduce the Riemannian metric $g = D^2 \Phi$ on $\mathbb{R}^d$ and study spectral properties of the metric-measure space $M=(\mathbb{R}^d, g, \mu)$. We prove, in particular, that $M$ admits a non-negative Bakry--Émery tensor provided both $V$ and $W$ are convex. If the target measure $\nu$ is the Lebesgue measure on a convex set $\Omega$ and $\mu$ is log-concave we prove that $M$ is a $CD(K,N)$ space. Applications of these results include some global dimension-free a priori estimates of $\| D^2 \Phi\|$. With the help of comparison techniques on Riemannian manifolds and probabilistic concentration arguments we proof some diameter estimates for $M$.
Citation: Alexander V. Kolesnikov. Hessian metrics, $CD(K,N)$-spaces, and optimal transportation of log-concave measures. Discrete & Continuous Dynamical Systems - A, 2014, 34 (4) : 1511-1532. doi: 10.3934/dcds.2014.34.1511
##### References:
 [1] I. J. Bakelman, "Convex Analysis and Nonlinear Geometric Elliptic Equations,", Springer-Verlag, (1994). doi: 10.1007/978-3-642-69881-1. Google Scholar [2] D. Bakry, Transformation de Riesz pour les semi-groupes symétrique. I. Étude de la dimension $1$,, in, 1123 (1985), 130. doi: 10.1007/BFb0075843. Google Scholar [3] D. Bakry and M. Émery, Diffusions hypercontractives,, in, 1123 (1985), 177. doi: 10.1007/BFb0075847. Google Scholar [4] V. I. Bogachev and A. V. Kolesnikov, On the Monge-Ampère equation in infinite dimensions,, Infin. Dimen. Anal. Quantum Probab. and Relat. Topics, 8 (2005), 547. doi: 10.1142/S0219025705002141. Google Scholar [5] V. I. Bogachev and A. V. Kolesnikov, Sobolev regularity for the Monge-Ampère equation in the Wiener space,, preprint, (). Google Scholar [6] L. A. Caffarelli, Interior $W^{2,p}$ estimates for solutions of the Monge-Ampère equation,, Ann. of Math. (2), 131 (1990), 135. doi: 10.2307/1971510. Google Scholar [7] L. A. Caffarelli and X. Cabré, "Fully Nonlinear Elliptic Equations,", American Mathematical Society Colloquium Publications, 43 (1995). Google Scholar [8] L. Caffarelli, L. Nirenberg and J. Spruck, The Dirichlet problem for nonlinear elliptic differential equations. I. Monge-Ampère equation,, CPAM, 37 (1984), 369. doi: 10.1002/cpa.3160370306. Google Scholar [9] E. Calabi, Improper affine hyperspheres of convex type and a generalization of a theorem by K. Jörgens,, Michigan Math. J., 5 (1958), 105. doi: 10.1307/mmj/1028998055. Google Scholar [10] S.-Y. Cheng and S.-T. Yau, The real Monge-Ampère equation and affine flat structures,, in, (1982), 339. Google Scholar [11] R. Eldan and B. Klartag, Approximately Gaussian marginals and the hyperplane conjecture,, in, 545 (2011), 55. doi: 10.1090/conm/545/10764. Google Scholar [12] D. Feyel and A. S. Üstünel, Monge-Kantorovich measure transportation and Monge-Ampère equation on Wiener space,, Prob. Theory and Related Fields, 128 (2004), 347. doi: 10.1007/s00440-003-0307-x. Google Scholar [13] D. Gilbarg and N. S. Trudinge, "Elliptic Partial Differential Equation of the Second Order,", Reprint of the 1998 edition, (1998). Google Scholar [14] M. Gromov, Convex sets and Kähler manifolds,, in, (1990), 1. Google Scholar [15] C. E. Gutièrrez, "The Monge-Ampère Equation,", Progress in Nonlinear Differential Equations and Their Applications, 44 (2001). doi: 10.1007/978-1-4612-0195-3. Google Scholar [16] B. Klartag, Poincaré inequalities and moment maps,, Annales de la Faculté des Sciences de Toulouse Sér. 6, 22 (2013), 1. doi: 10.5802/afst.1366. Google Scholar [17] A. V. Kolesnikov, Global Hölder estimates for optimal transportation,, Mat. Zametki, 88 (2010), 708. doi: 10.1134/S0001434610110076. Google Scholar [18] A. V. Kolesnikov, On Sobolev regularity of mass transport and transportation inequalities,, Theory Probab. Appl., 57 (2012), 243. doi: 10.1137/S0040585X97985947. Google Scholar [19] A. V. Kolesnikov, Convexity inequalities and optimal transport of infinite-dimensional measures,, J. Math. Pures Appl. (9), 83 (2004), 1373. doi: 10.1016/j.matpur.2004.03.005. Google Scholar [20] A. V. Kolesnikov, Mass transportation and contractions,, MIPT Proc., 2 (2010), 90. Google Scholar [21] N. V. Krylov, Fully nonlinear second order elliptic equations: Recent developments,, Ann. Scuola Norm. Sup. Pisa Cl. Sci (4), 25 (1997), 569. Google Scholar [22] M. Ledoux, Concentration of measure and logarithmic Sobolev inequality,, in, 1709 (1999), 120. doi: 10.1007/BFb0096511. Google Scholar [23] E. Milman, Isoperimetric and concentration inequalities: Equivalence under curvature lower bound,, Duke Math. J., 154 (2010), 207. doi: 10.1215/00127094-2010-038. Google Scholar [24] E. Milman, On the role of convexity in isoperimetry, spectral gap and concentration,, Invent. Math., 177 (2009), 1. doi: 10.1007/s00222-009-0175-9. Google Scholar [25] L. E. Payne and H. F. Weinberger, An optimal Poincaré inequality for convex domains,, Arch. Rational Mech. Anal., 5 (1960), 286. doi: 10.1007/BF00252910. Google Scholar [26] {A. V. Pogorelov}, "Monge-Ampère Equations of Elliptic Type,", Noordhoff, (1964). Google Scholar [27] H. Shima, "The Geometry of Hessian Structures,", World Scientific Publishing Co. Pte. Ltd., (2007). doi: 10.1142/9789812707536. Google Scholar [28] N. S. Trudinger and X.-L. Wang, The Monge-Ampère equation and its geometric applications, in, 7 (2008), 467. Google Scholar [29] C. Villani, "Optimal Transport. Old and New,", Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 338 (2009). doi: 10.1007/978-3-540-71050-9. Google Scholar

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##### References:
 [1] I. J. Bakelman, "Convex Analysis and Nonlinear Geometric Elliptic Equations,", Springer-Verlag, (1994). doi: 10.1007/978-3-642-69881-1. Google Scholar [2] D. Bakry, Transformation de Riesz pour les semi-groupes symétrique. I. Étude de la dimension $1$,, in, 1123 (1985), 130. doi: 10.1007/BFb0075843. Google Scholar [3] D. Bakry and M. Émery, Diffusions hypercontractives,, in, 1123 (1985), 177. doi: 10.1007/BFb0075847. Google Scholar [4] V. I. Bogachev and A. V. Kolesnikov, On the Monge-Ampère equation in infinite dimensions,, Infin. Dimen. Anal. Quantum Probab. and Relat. Topics, 8 (2005), 547. doi: 10.1142/S0219025705002141. Google Scholar [5] V. I. Bogachev and A. V. Kolesnikov, Sobolev regularity for the Monge-Ampère equation in the Wiener space,, preprint, (). Google Scholar [6] L. A. Caffarelli, Interior $W^{2,p}$ estimates for solutions of the Monge-Ampère equation,, Ann. of Math. (2), 131 (1990), 135. doi: 10.2307/1971510. Google Scholar [7] L. A. Caffarelli and X. Cabré, "Fully Nonlinear Elliptic Equations,", American Mathematical Society Colloquium Publications, 43 (1995). Google Scholar [8] L. Caffarelli, L. Nirenberg and J. Spruck, The Dirichlet problem for nonlinear elliptic differential equations. I. Monge-Ampère equation,, CPAM, 37 (1984), 369. doi: 10.1002/cpa.3160370306. Google Scholar [9] E. Calabi, Improper affine hyperspheres of convex type and a generalization of a theorem by K. Jörgens,, Michigan Math. J., 5 (1958), 105. doi: 10.1307/mmj/1028998055. Google Scholar [10] S.-Y. Cheng and S.-T. Yau, The real Monge-Ampère equation and affine flat structures,, in, (1982), 339. Google Scholar [11] R. Eldan and B. Klartag, Approximately Gaussian marginals and the hyperplane conjecture,, in, 545 (2011), 55. doi: 10.1090/conm/545/10764. Google Scholar [12] D. Feyel and A. S. Üstünel, Monge-Kantorovich measure transportation and Monge-Ampère equation on Wiener space,, Prob. Theory and Related Fields, 128 (2004), 347. doi: 10.1007/s00440-003-0307-x. Google Scholar [13] D. Gilbarg and N. S. Trudinge, "Elliptic Partial Differential Equation of the Second Order,", Reprint of the 1998 edition, (1998). Google Scholar [14] M. Gromov, Convex sets and Kähler manifolds,, in, (1990), 1. Google Scholar [15] C. E. Gutièrrez, "The Monge-Ampère Equation,", Progress in Nonlinear Differential Equations and Their Applications, 44 (2001). doi: 10.1007/978-1-4612-0195-3. Google Scholar [16] B. Klartag, Poincaré inequalities and moment maps,, Annales de la Faculté des Sciences de Toulouse Sér. 6, 22 (2013), 1. doi: 10.5802/afst.1366. Google Scholar [17] A. V. Kolesnikov, Global Hölder estimates for optimal transportation,, Mat. Zametki, 88 (2010), 708. doi: 10.1134/S0001434610110076. Google Scholar [18] A. V. Kolesnikov, On Sobolev regularity of mass transport and transportation inequalities,, Theory Probab. Appl., 57 (2012), 243. doi: 10.1137/S0040585X97985947. Google Scholar [19] A. V. Kolesnikov, Convexity inequalities and optimal transport of infinite-dimensional measures,, J. Math. Pures Appl. (9), 83 (2004), 1373. doi: 10.1016/j.matpur.2004.03.005. Google Scholar [20] A. V. Kolesnikov, Mass transportation and contractions,, MIPT Proc., 2 (2010), 90. Google Scholar [21] N. V. Krylov, Fully nonlinear second order elliptic equations: Recent developments,, Ann. Scuola Norm. Sup. Pisa Cl. Sci (4), 25 (1997), 569. Google Scholar [22] M. Ledoux, Concentration of measure and logarithmic Sobolev inequality,, in, 1709 (1999), 120. doi: 10.1007/BFb0096511. Google Scholar [23] E. Milman, Isoperimetric and concentration inequalities: Equivalence under curvature lower bound,, Duke Math. J., 154 (2010), 207. doi: 10.1215/00127094-2010-038. Google Scholar [24] E. Milman, On the role of convexity in isoperimetry, spectral gap and concentration,, Invent. Math., 177 (2009), 1. doi: 10.1007/s00222-009-0175-9. Google Scholar [25] L. E. Payne and H. F. Weinberger, An optimal Poincaré inequality for convex domains,, Arch. Rational Mech. Anal., 5 (1960), 286. doi: 10.1007/BF00252910. Google Scholar [26] {A. V. Pogorelov}, "Monge-Ampère Equations of Elliptic Type,", Noordhoff, (1964). Google Scholar [27] H. Shima, "The Geometry of Hessian Structures,", World Scientific Publishing Co. Pte. Ltd., (2007). doi: 10.1142/9789812707536. Google Scholar [28] N. S. Trudinger and X.-L. Wang, The Monge-Ampère equation and its geometric applications, in, 7 (2008), 467. Google Scholar [29] C. Villani, "Optimal Transport. Old and New,", Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 338 (2009). doi: 10.1007/978-3-540-71050-9. Google Scholar
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