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Mass Transportation on surfaces

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73 pages
Mass Transportation on surfaces Ludovic Rifford Universite de Nice - Sophia Antipolis Ludovic Rifford Mass Transportation on surfaces

  • lebesgue measure

  • optimal transport map

  • any measurable map

  • monge quadratic

  • quadratic cost


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Mass
Transportation on surfaces
Ludovic Rifford
Universit´edeNice-SophiaAntipolis
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measurable
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,
B
onsutiones
Letµ0andµ1beprobability measures with compact supportinRn. We calltransport mapfromµ0toµ1any measurable mapT:RnRnsuch thatT]µ0=µ1, that is µ1(B) =µ0T1(B),BmeasurableRn.
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Monge quadratic problem: Study of transport maps T:RnRnwhich minimize thequadratictransport cost Z
se
dµ0(x).
R
|T(x)− |2 x n
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Monge quadratic problem of transport maps: Study T:RnRnwhich minimize thequadratictransport cost ZRn|T(x)x|2dµ0(x).
Theorem (Brenier ’91) Assume thatµ0is absolutely continuous with respect to the Lebesgue measure. Then there exists a unique optimal transport map for the quadratic cost fromµ0toµ1.
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T(x) =rψ(x)µ0a.e. xRn.
Theorem (Brenier ’91) Assume thatµ0is absolutely continuous with respect to the Lebesgue measure. Then there exists a unique optimal transport map for the quadratic cost fromµ0toµ1. There is a convex functionψ:MRsuch that
Monge quadratic problem: Study of transport maps T:RnRnwhich minimize thequadratictransport cost ZR|T(x)x|2dµ0(x). n
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Theorem (Brenier ’91) Assume thatµ0is absolutely continuous with respect to the Lebesgue measure. Then there exists a unique optimal transport map for the quadratic cost fromµ0toµ1. There is a convex functionψ:MRsuch that
Monge quadratic problem: Study of transport maps T:RnRnwhich minimize thequadratictransport cost ZRn|T(x)x|2dµ0(x).
Regularity ?
T(x) =rψ(x)µ0a.e. xRn.
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Contre-exemple
trivial
Ludovic
Rifford
Mass
Transp
ortation
on
surfaces
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