# Optimal transportation for the determinant

Guillaume Carlier; Bruno Nazaret

ESAIM: Control, Optimisation and Calculus of Variations (2008)

- Volume: 14, Issue: 4, page 678-698
- ISSN: 1292-8119

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topCarlier, Guillaume, and Nazaret, Bruno. "Optimal transportation for the determinant." ESAIM: Control, Optimisation and Calculus of Variations 14.4 (2008): 678-698. <http://eudml.org/doc/250318>.

@article{Carlier2008,

abstract = {
Among $\{\mathbb R\}^3$-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.
},

author = {Carlier, Guillaume, Nazaret, Bruno},

journal = {ESAIM: Control, Optimisation and Calculus of Variations},

keywords = {Optimal transportation; multi-marginals problems; determinant; disintegrations; optimal transportation},

language = {eng},

month = {1},

number = {4},

pages = {678-698},

publisher = {EDP Sciences},

title = {Optimal transportation for the determinant},

url = {http://eudml.org/doc/250318},

volume = {14},

year = {2008},

}

TY - JOUR

AU - Carlier, Guillaume

AU - Nazaret, Bruno

TI - Optimal transportation for the determinant

JO - ESAIM: Control, Optimisation and Calculus of Variations

DA - 2008/1//

PB - EDP Sciences

VL - 14

IS - 4

SP - 678

EP - 698

AB -
Among ${\mathbb R}^3$-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.

LA - eng

KW - Optimal transportation; multi-marginals problems; determinant; disintegrations; optimal transportation

UR - http://eudml.org/doc/250318

ER -

## References

top- Y. Brenier, Polar factorization and monotone rearrangements of vector valued functions. Comm. Pure Appl. Math.44 (1991) 375–417.
- B. Dacorogna, Direct methods in the calculus of variations, Applied Mathematical Sciences78. Springer-Verlag, Berlin (1989).
- I. Ekeland, A duality theorem for some non-convex functions of matrices. Ric. Mat.55 (2006) 1–12.
- I. Ekeland and R. Temam, Convex Analysis and Variational Problems, in Classics in Mathematics, Society for Industrial and Applied Mathematics, Philadelphia (1999).
- W. Gangbo and A. Święch, Optimal maps for the multidimensional Monge-Kantorovich problem. Comm. Pure Appl. Math.51 (1998) 23–45.
- W. Gangbo and R.J. McCann, The geometry of optimal transportation. Acta Math. 177 (1996) 113–161.
- S.T. Rachev and L. Rüschendorf, Mass Transportation Problems. Vol. I: Theory; Vol. II: Applications. Springer-Verlag (1998).
- C. Villani, Topics in optimal transportation, Graduate Studies in Mathematics58. American Mathematical Society, Providence, RI (2003).

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