Rectifiability of Optimal Transportation Plans
Canadian journal of mathematics, Tome 64 (2012) no. 4, pp. 924-934
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The regularity of solutions to optimal transportation problems has become a hot topic in current research. It is well known by now that the optimal measure may not be concentrated on the graph of a continuous mapping unless both the transportation cost and the masses transported satisfy very restrictive hypotheses (including sign conditions on the mixed fourth-order derivatives of the cost function). The purpose of this note is to show that in spite of this, the optimal measure is supported on a Lipschitz manifold, provided only that 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.
McCann, Robert J.; Pass, Brendan; Warren, Micah. Rectifiability of Optimal Transportation Plans. Canadian journal of mathematics, Tome 64 (2012) no. 4, pp. 924-934. doi: 10.4153/CJM-2011-080-6
@article{10_4153_CJM_2011_080_6,
author = {McCann, Robert J. and Pass, Brendan and Warren, Micah},
title = {Rectifiability of {Optimal} {Transportation} {Plans}},
journal = {Canadian journal of mathematics},
pages = {924--934},
year = {2012},
volume = {64},
number = {4},
doi = {10.4153/CJM-2011-080-6},
url = {http://geodesic.mathdoc.fr/articles/10.4153/CJM-2011-080-6/}
}
TY - JOUR AU - McCann, Robert J. AU - Pass, Brendan AU - Warren, Micah TI - Rectifiability of Optimal Transportation Plans JO - Canadian journal of mathematics PY - 2012 SP - 924 EP - 934 VL - 64 IS - 4 UR - http://geodesic.mathdoc.fr/articles/10.4153/CJM-2011-080-6/ DO - 10.4153/CJM-2011-080-6 ID - 10_4153_CJM_2011_080_6 ER -
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