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 $\left(X,Y,Z\right)$ having fixed marginal probability laws, what is the best way to jointly draw $\left(X,Y,Z\right)$ in such a way that the simplex generated by $\left(X,Y,Z\right)$ 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 : https://doi.org/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},
zbl = {1160.49015},
mrnumber = {2451790},
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
DA  - 2008///
SP  - 678
EP  - 698
VL  - 14
IS  - 4
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/cocv:2008006/
UR  - https://zbmath.org/?q=an%3A1160.49015
UR  - https://www.ams.org/mathscinet-getitem?mr=2451790
UR  - https://doi.org/10.1051/cocv:2008006
DO  - 10.1051/cocv:2008006
LA  - en
ID  - COCV_2008__14_4_678_0
ER  - 
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 1100809 | Zbl 0738.46011

[2] B. Dacorogna, Direct methods in the calculus of variations, Applied Mathematical Sciences 78. Springer-Verlag, Berlin (1989). | MR 990890 | Zbl 0703.49001

[3] I. Ekeland, A duality theorem for some non-convex functions of matrices. Ric. Mat. 55 (2006) 1-12. | MR 2248159 | Zbl 1220.15012

[4] I. Ekeland and R. Temam, Convex Analysis and Variational Problems, in Classics in Mathematics, Society for Industrial and Applied Mathematics, Philadelphia (1999). | MR 1727362 | Zbl 0939.49002

[5] W. Gangbo and A. Świȩch, Optimal maps for the multidimensional Monge-Kantorovich problem. Comm. Pure Appl. Math. 51 (1998) 23-45. | MR 1486630 | Zbl 0889.49030

[6] W. Gangbo and R.J. Mccann, The geometry of optimal transportation. Acta Math. 177 (1996) 113-161. | MR 1440931 | Zbl 0887.49017

[7] S.T. Rachev and L. Rüschendorf, Mass Transportation Problems. Vol. I: Theory; Vol. II: Applications. Springer-Verlag (1998). | MR 1619170 | Zbl 0990.60500

[8] C. Villani, Topics in optimal transportation, Graduate Studies in Mathematics 58. American Mathematical Society, Providence, RI (2003). | MR 1964483 | Zbl 1106.90001

Cité par Sources :