Brendan Pass
Current Organizations / Institutes
The organizations/institutes from the scholar's latest article
Brendan Pass has a total of 2 co-authors. Here are the top co-authors by number of shared publications:
- Micah Warren
Princeton University
1 shared publication - Robert J. McCann
Department of Mathematics, University of Toronto, Ontario, Canada
1 shared publication
Latest Publications
- 30 March 2011
- Published by Springer Nature in Calculus of Variations and Partial Differential Equations
- Vol. 43 (3-4) , 529-536
On the local structure of optimal measures in the multi-marginal optimal transportation problem
- 30 March 2011
- Published by Springer Nature in Calculus of Variations and Partial Differential Equations
- Vol. 43 (3-4) , 529-536
On the local structure of optimal measures in the multi-marginal optimal transportation problem
- Open Access
Regularity of optimal transportation between spaces with different dimensions
We study the regularity of solutions to an optimal transportation problem where the dimension of the source is larger than that of the target. We demonstrate that if the target is $c$-convex, then the source has a canonical foliation whose co-dimension is equal to the dimension of the target and the problem reduces to an optimal transportation problem between spaces with equal dimensions. If the $c$-convexity condition fails, we do not expect regularity for arbitrary smooth marginals, but, in the case where the source is 2-dimensional and the target is 1 dimensional, we identify sufficient conditions on the marginals and cost to ensure that the optimal map is continuous.
Open AccessRegularity of optimal transportation between spaces with different dimensions
We study the regularity of solutions to an optimal transportation problem where the dimension of the source is larger than that of the target. We demonstrate that if the target is $c$-convex, then the source has a canonical foliation whose co-dimension is equal to the dimension of the target and the problem reduces to an optimal transportation problem between spaces with equal dimensions. If the $c$-convexity condition fails, we do not expect regularity for arbitrary smooth marginals, but, in the case where the source is 2-dimensional and the target is 1 dimensional, we identify sufficient conditions on the marginals and cost to ensure that the optimal map is continuous.
- Open Access
Uniqueness and Monge solutions in the multi-marginal optimal transportation problem
We study a multi-marginal optimal transportation problem. Under certain conditions on the cost function and the first marginal, we prove that the solution to the relaxed, Kantorovich version of the problem induces a solution to the Monge problem and that the solutions to both problems are unique.
Open AccessUniqueness and Monge solutions in the multi-marginal optimal transportation problem
We study a multi-marginal optimal transportation problem. Under certain conditions on the cost function and the first marginal, we prove that the solution to the relaxed, Kantorovich version of the problem induces a solution to the Monge problem and that the solutions to both problems are unique.
- Open Access
On the local structure of optimal measures in the multi-marginal optimal transportation problem
We consider an optimal transportation problem with more than two marginals. We use a family of semi-Riemannian metrics derived from the mixed, second order partial derivatives of the cost function to provide upper bounds for the dimension of the support of the solution.
Open AccessOn the local structure of optimal measures in the multi-marginal optimal transportation problem
We consider an optimal transportation problem with more than two marginals. We use a family of semi-Riemannian metrics derived from the mixed, second order partial derivatives of the cost function to provide upper bounds for the dimension of the support of the solution.
- Open Access
Rectifiability of Optimal Transportation Plans
The purpose of this note is to show that the solution to the Kantorovich optimal transportation problem is supported on a Lipschitz manifold, provided the cost is $C^{2}$ with non-singular mixed second derivative. We use this result to provide a simple proof that solutions to Monge's optimal transportation problem satisfy a change of variables equation almost everywhere.Open AccessRectifiability of Optimal Transportation Plans
The purpose of this note is to show that the solution to the Kantorovich optimal transportation problem is supported on a Lipschitz manifold, provided the cost is $C^{2}$ with non-singular mixed second derivative. We use this result to provide a simple proof that solutions to Monge's optimal transportation problem satisfy a change of variables equation almost everywhere.