Séminaire Bourbaki
(2009)
Author:
Alessio Figalli
Abstract:
The issue of regularity of optimal transport maps in the case
``cost=squared distance'' on R^n
was solved by Caffarelli in the 1990s. However, a major open problem in
the theory was the question of regularity
for more general cost functions, or for the case ``cost=squared distance''
on a Riemannian manifold.
A breakthrough to this problem has been achieved by Ma-Trudinger-Wang and Loeper, who found a necessary
and sufficient condition on the cost function in order to ensure the
regularity of the optimal map.
In the special case ``cost=squared distance'' on a Riemannian manifold,
this condition corresponds to the non-negativity of a new curvature tensor
on the manifold, which implies strong geometric consequences on the
geometry of the manifold and on the structure of its cut-locus.![[PS]](/style/ps.png)
![[PDF]](/style/pdf.png)
[BibTeX Entry]
Available Files:
abstract.tex (abstract.ps, abstract.pdf)
Exp_1009_A_Figalli.pdf
