no. 4
Optimal transportation for the determinant
ESAIM: Control, Optimisation and Calculus of Variations, Tome 14 (2008) no. 4, pp. 678-698.

Among ℝ 3 -valued triples of random vectors (X,Y,Z) having fixed marginal probability laws, what is the best way to jointly draw (X,Y,Z) in such a way that the simplex generated by (X,Y,Z) has maximal average volume? Motivated by this simple question, we study optimal transportation problems with several marginals when the objective function is the determinant or its absolute value.

DOI : 10.1051/cocv:2008006
Classification : 28-99, 49-99
Mots clés : optimal transportation, multi-marginals problems, determinant, disintegrations 
@article{COCV_2008__14_4_678_0,
     author = {Carlier, Guillaume and Nazaret, Bruno},
     title = {Optimal transportation for the determinant},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {678--698},
     publisher = {EDP-Sciences},
     volume = {14},
     number = {4},
     year = {2008},
     doi = {10.1051/cocv:2008006},
     mrnumber = {2451790},
     zbl = {1160.49015},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/cocv:2008006/}
}
TY  - JOUR
AU  - Carlier, Guillaume
AU  - Nazaret, Bruno
TI  - Optimal transportation for the determinant
JO  - ESAIM: Control, Optimisation and Calculus of Variations
PY  - 2008
SP  - 678
EP  - 698
VL  - 14
IS  - 4
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/cocv:2008006/
DO  - 10.1051/cocv:2008006
LA  - en
ID  - COCV_2008__14_4_678_0
ER  - 
%0 Journal Article
%A Carlier, Guillaume
%A Nazaret, Bruno
%T Optimal transportation for the determinant
%J ESAIM: Control, Optimisation and Calculus of Variations
%D 2008
%P 678-698
%V 14
%N 4
%I EDP-Sciences
%U http://www.numdam.org/articles/10.1051/cocv:2008006/
%R 10.1051/cocv:2008006
%G en
%F COCV_2008__14_4_678_0
Carlier, Guillaume; Nazaret, Bruno. Optimal transportation for the determinant. ESAIM: Control, Optimisation and Calculus of Variations, Tome 14 (2008) no. 4, pp. 678-698. doi : 10.1051/cocv:2008006. http://www.numdam.org/articles/10.1051/cocv:2008006/

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

[2] B. Dacorogna, Direct methods in the calculus of variations, Applied Mathematical Sciences 78. Springer-Verlag, Berlin (1989). | MR | Zbl

[3] I. Ekeland, A duality theorem for some non-convex functions of matrices. Ric. Mat. 55 (2006) 1-12. | MR | Zbl

[4] I. Ekeland and R. Temam, Convex Analysis and Variational Problems, in Classics in Mathematics, Society for Industrial and Applied Mathematics, Philadelphia (1999). | MR | Zbl

[5] W. Gangbo and A. Świȩch, Optimal maps for the multidimensional Monge-Kantorovich problem. Comm. Pure Appl. Math. 51 (1998) 23-45. | MR | Zbl

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

[7] S.T. Rachev and L. Rüschendorf, Mass Transportation Problems. Vol. I: Theory; Vol. II: Applications. Springer-Verlag (1998). | MR | Zbl

[8] C. Villani, Topics in optimal transportation, Graduate Studies in Mathematics 58. American Mathematical Society, Providence, RI (2003). | MR | Zbl

Cité par Sources :