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

Abstract

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Among 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.

How to cite

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Carlier, 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

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  2. B. Dacorogna, Direct methods in the calculus of variations, Applied Mathematical Sciences78. Springer-Verlag, Berlin (1989).  Zbl0703.49001
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  5. W. Gangbo and A. Święch, Optimal maps for the multidimensional Monge-Kantorovich problem. Comm. Pure Appl. Math.51 (1998) 23–45.  Zbl0889.49030
  6. W. Gangbo and R.J. McCann, The geometry of optimal transportation. Acta Math. 177 (1996) 113–161.  Zbl0887.49017
  7. S.T. Rachev and L. Rüschendorf, Mass Transportation Problems. Vol. I: Theory; Vol. II: Applications. Springer-Verlag (1998).  Zbl0990.60500
  8. C. Villani, Topics in optimal transportation, Graduate Studies in Mathematics58. American Mathematical Society, Providence, RI (2003).  Zbl1106.90001

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