On the hessian of the optimal transport potential
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 6 (2007) no. 3, pp. 441-456.

We study the optimal solution of the Monge-Kantorovich mass transport problem between measures whose density functions are convolution with a gaussian measure and a log-concave perturbation of a different gaussian measure. Under certain conditions we prove bounds for the Hessian of the optimal transport potential. This extends and generalises a result of Caffarelli. We also show how this result fits into the scheme of Barthe to prove Brascamp-Lieb inequalities and thus prove a new generalised Reverse Brascamp-Lieb inequality.

Classification : 49Q20, 52A40, 44A35 
@article{ASNSP_2007_5_6_3_441_0,
     author = {Valdimarsson, Stef\'an Ingi},
     title = {On the hessian of the optimal transport potential},
     journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
     pages = {441--456},
     publisher = {Scuola Normale Superiore, Pisa},
     volume = {Ser. 5, 6},
     number = {3},
     year = {2007},
     mrnumber = {2370268},
     zbl = {1170.49038},
     language = {en},
     url = {http://archive.numdam.org/item/ASNSP_2007_5_6_3_441_0/}
}
TY  - JOUR
AU  - Valdimarsson, Stefán Ingi
TI  - On the hessian of the optimal transport potential
JO  - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
PY  - 2007
SP  - 441
EP  - 456
VL  - 6
IS  - 3
PB  - Scuola Normale Superiore, Pisa
UR  - http://archive.numdam.org/item/ASNSP_2007_5_6_3_441_0/
LA  - en
ID  - ASNSP_2007_5_6_3_441_0
ER  - 
%0 Journal Article
%A Valdimarsson, Stefán Ingi
%T On the hessian of the optimal transport potential
%J Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
%D 2007
%P 441-456
%V 6
%N 3
%I Scuola Normale Superiore, Pisa
%U http://archive.numdam.org/item/ASNSP_2007_5_6_3_441_0/
%G en
%F ASNSP_2007_5_6_3_441_0
Valdimarsson, Stefán Ingi. On the hessian of the optimal transport potential. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 6 (2007) no. 3, pp. 441-456. http://archive.numdam.org/item/ASNSP_2007_5_6_3_441_0/

[1] S. Alesker, S. Dar, and V. Milman, A remarkable measure preserving diffeomorphism between two convex bodies in ℝ n , Geom. Dedicata 74 (1999), 201-212. | MR | Zbl

[2] F. Barthe, On a reverse form of the Brascamp-Lieb inequality, Invent. Math. 134 (1998), 335-361. | MR | Zbl

[3] J. Bennett, A. Carbery, M. Christ and T. Tao, The Brascamp-Lieb inequalities: finiteness, structure and extremals, Geom. Funct. Anal., to appear. | MR | Zbl

[4] Y. Brenier, Décomposition polaire et réarrangement monotone des champs de vecteurs, C. R. Acad. Sci. Paris Sér. I Math. 305 (1987), 805-808. | MR | Zbl

[5] Y. Brenier, Polar factorization and monotone rearrangement of vector-valued functions, Comm. Pure Appl. Math. 44 (1991), 375-417. | MR | Zbl

[6] L. A. Caffarelli, Monotonicity properties of optimal transportation and the FKG and related inequalities, Comm. Math. Phys. 214 (2000), 547-563. | MR | Zbl

[7] L. A. Caffarelli, Erratum: Monotonicity of optimal transportation and the FKG and related inequalities, Comm. Math. Phys. 214 (2000), 547-563; Comm. Math. Phys. 225 (2002), 449-450. | Zbl

[8] W. Gangbo and R. J. Mccann, The geometry of optimal transportation, Acta Math. 177 (1996), 113-161. | MR | Zbl

[9] R. A. Horn and C. R. Johnson, “Matrix Analysis”, Cambridge University Press, Cambridge, 1985. | MR | Zbl

[10] E. H. Lieb, Gaussian kernels have only Gaussian maximizers, Invent. Math. 102 (1990), 179-208. | MR | Zbl

[11] C. Villani, “Topics in Optimal Transportation”, Graduate Studies in Mathematics, Vol. 58, American Mathematical Society, Providence, RI, 2003. | MR | Zbl

[12] C. Villani, “Optimal Transport, Old and New”, 2007, July 18, preprint. | MR | Zbl