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multi-marginals problems

ESAIM: Control, Optimisation and Calculus of Variations (1)

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Optimal transportation for the determinant
Guillaume Carlier, Bruno Nazaret
Journal:

ESAIM: Control, Optimisation and Calculus of Variations / Volume 14 / Issue 4 / October 2008

Published online by Cambridge University Press:

18 January 2008, pp. 678-698

Print publication:

October 2008
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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.