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Optimal Transport on Surfaces

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41 pages
Optimal Transport on Surfaces Ludovic Rifford Universite de Nice - Sophia Antipolis Ludovic Rifford Topics on Optimal Transport (IRMA, 16-17 septembre 2010)

  • riemannian surface

  • optimal transport

  • µ0

  • mesurable map

  • geodesic distance

  • universite de nice

  • compact connected


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Ludovic Rifford
Universite´deNice-SophiaAntipolis
Optimal Transport on Surfaces
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OptimaltiRnonamesnartropldfosanninimaeoegthdgbytenoDeddnaeneqehtrdausideiscdnctanMeo,b)cyx(y,:)1=d2aticcostc:M×M[0oBowtnevrpnailer2y)x,g(GiM.yx,1µnoµs,0adem,Mnilitobabsureymea]µgTµ10=isatinfy:TpasMMarusmelbborelian1(B),B)Bµ=0Ti(e.µ.(10(dµ))(x,T(xMcgZniziminimdna,)MmalTOpticsonTopiodrciiRduvo)xL.
a
be
compact
smooth
Riemannian
connected
)
Let (M,g) surface.
portransA,16(IRMpeet1-s70201bmer
ponsonrtemRinianamnaofinsdlpOitamtlarno,M,01µemusnaditymabilresµeasueroBowtnborpnailveGi)B,BoberilnaM.e.µ1(B)=µ0T1(yfsiTgni=0µ]i(1µblraapemMT:atMslTratimaonOppicsdroTiRovociL.dux)0(dµ))(x,T(xMcZgniziminimdna,)
12 c(x,y) := 2dg(x,y)
Let (M,g) be a smooth compact connected Riemannian surface. Denote bydgthe geodesic distance onMand define the quadratic costc:M×M[0,) by
x,yM.
20re)10ep7smbte,AMR1-61opsnI(tr
psnartlamitpOrootRneiamnnaimnanifoldsimptnOsospanTraliRcivodcipoTdrobre2ptem
Given two Borelian probability measuresµ0, µ1onM, find a mesurable mapT:MMsatisfying T]µ0=µ1(i.e.µ1(B) =µ0T1(B),BborelianM),
c(x,y21=:)dg(x,y)2x,yM.
Let (M,g) be a smooth compact connected Riemannian surface. Denote bydgthe geodesic distance onMand define the quadratic costc:M×M[0,) by
010)RIAMro(t71es1,-6Lu
Z
c(x,T(x))dµ0(x). M
and minimizing