//img.uscri.be/pth/81a20c9425bac03eb0342c78b9c78b83f68d6a51
Cet ouvrage fait partie de la bibliothèque YouScribe
Obtenez un accès à la bibliothèque pour le lire en ligne
En savoir plus

REGULARITY OF OPTIMAL TRANSPORT IN CURVED GEOMETRY: THE NONFOCAL CASE

55 pages
REGULARITY OF OPTIMAL TRANSPORT IN CURVED GEOMETRY: THE NONFOCAL CASE G. LOEPER AND C. VILLANI Abstract. We explore some geometric and analytic consequences of a curvature condition introduced by Ma, Trudinger and Wang in relation to the smoothness of optimal transport in curved geometry. We discuss a conjecture according to which a strict version of the Ma–Trudinger–Wang condition is sufficient to prove regu- larity of optimal transport on a Riemannian manifold. We prove this conjecture under a somewhat restrictive additional assumption of nonfocality; at the same time, we establish the striking geometric property that the tangent cut locus is the boundary of a convex set. Partial extensions are presented to the case when there is no “pure focalization” on the tangent cut locus. Contents 1. Introduction 2 2. Various forms of the Ma–Trudinger–Wang condition 9 3. Metric consequences of the Ma–Trudinger–Wang condition 11 4. Uniform regularity 15 5. Convexity of injectivity domains 19 6. From c-convexity to C1 regularity 30 7. Stay-away property 33 8. Holder continuity of optimal transport 36 9. Final comments and open problems 39 Appendix A. Uniform convexity 40 Appendix B. Semiconvexity 42 Appendix C. Differential structure of the tangent cut locus 43 Appendix D. A counterexample 52 References 53 1

  • riemannian manifold

  • transport

  • any ? ?

  • question nontrivial

  • collects various sufficient

  • optimal transport

  • mtw condition

  • topology —

  • various forms


Voir plus Voir moins

REGULARITYOFOPTIMALTRANSPORTINCURVED
GEOMETRY:THENONFOCALCASE
G.LOEPERANDC.VILLANI
Abstract.
Weexploresomegeometricandanalyticconsequencesofacurvature
conditionintroducedbyMa,TrudingerandWanginrelationtothesmoothnessof
optimaltransportincurvedgeometry.Wediscussaconjectureaccordingtowhich
astrictversionoftheMa–Trudinger–Wangconditionissufficienttoproveregu-
larityofoptimaltransportonaRiemannianmanifold.Weprovethisconjecture
underasomewhatrestrictiveadditionalassumptionofnonfocality;atthesame
time,weestablishthestrikinggeometricpropertythatthetangentcutlocusis
theboundaryofaconvexset.Partialextensionsarepresentedtothecasewhen
thereisno“purefocalization”onthetangentcutlocus.

Contents
1.Introduction
2.VariousformsoftheMa–Trudinger–Wangcondition
3.MetricconsequencesoftheMa–Trudinger–Wangcondition
4.Uniformregularity
5.Convexityofinjectivitydomains
6.From
c
-convexityto
C
1
regularity
7.Stay-awayproperty
8.Ho¨ldercontinuityofoptimaltransport
9.Finalcommentsandopenproblems
AppendixA.Uniformconvexity
AppendixB.Semiconvexity
AppendixC.Differentialstructureofthetangentcutlocus
AppendixD.Acounterexample
References

1

29115191033363930424342535

2

G.LOEPERANDC.VILLANI

1.
Introduction
Thispaperhastwosides:ontheonehand,itisaworkonthesmoothnessof
optimaltransport;ontheotherhand,itisaworkonthestructureofthecutlocus.
Thelattercouldbediscussedindependentlyoftheformer,butsincetheinitial
motivationwasinoptimaltransporttheory,andsincebothfeaturesareintimately
entangled,weshallpresentbothproblematicstogether.Ourintroductionisreduced
totheminimumthatthereadershouldknowtounderstandthepaper;butmuch
moreinformationcanbefoundinthebooks[29,30];especially[30,Chapter12]is
alongandself-containedintroductiontotheregularityofoptimaltransport.
1.1.
Regularityofoptimaltransport:backgroundandmainresult.
After
Caffarelli[2,3,4]andUrbas[28]studiedthesmoothnessofoptimaltransportmaps
forthequadraticcostfunctionin
R
n
,theproblemnaturallyarosetoextendthese
resultstomoregeneralcostfunctions[29,Section4.3].Inthispaper,weshallonly
considertheimportantcasewhenthecostisthesquaredgeodesicdistanceona
Riemannianmanifold
M
;thiscostfunction,firststudiedbyMcCann[24],hasmany
applicationsinRiemanniangeometry[30,PartII].
Therewasalmostnoprogressonthesmoothnessissuebeforetheintroductionof
theMa–Trudinger–Wangtensor[22].Let
M
beaRiemannianmanifold,whichas
intherestofthispaperwillimplicitlybeassumedtobesmooth,connectedand
complete.Let
TM
=

(
{
x

T
x
M
)standforthetangentbundleover
M
,and
letcut(
M
)=

(
{
x

cut(
x
))denotethecutlocusof
M
.TheMa–Trudinger–
Wang(MTW)tensor
S
canbedefinedon
T
(
M
×
M
\
cut(
M
))asfollows[30,
Definition12.26].Let(
x,y
)

M
×
M
\
cut(
M
),takecoordinatesystems(
x
i
)
1

i

n
,
(
y
j
)
1

j

n
around
x
and
y
respectively;set
c
(
x

,y

)=
d
(
x

,y

)
2
/
2,where
d
isthe
geodesicdistanceon
M
,andnotethat
c
is
C

around(
x,y
).Write
c
i
(resp.
c
,j
)
forthepartialderivativewithrespectto
x
i
(resp.
y
j
),evaluatedat(
x,y
);
c
i,j
for
themixedsecondderivativewithrespectto
x
i
and
y
j
,etc.;andwrite(
c
i,j
)forthe
componentsoftheinverseof(
c
i,j
),alwaysevaluatedat(
x,y
).Thenforany
ξ

T
x
M
,
η

T
y
M
,
3X(1.1)
S
(
x,y
)

(
ξ,η
):=
c
ij,r
c
r,s
c
s,kℓ

c
ij,kℓ
ξ
i
ξ
j
η
k
η

.
2
ijkℓrs
AccordingtoLoeper[20],KimandMcCann[16],thisformuladefinesacovariant
tensor.Moreover,asnotedin[20],if
ξ
and
η
areorthogonalunitvectorsin
T
x
M
,
then
S
(
x,x
)

(
ξ,η
)coincideswiththesectionalcurvatureat
x
alongtheplane
generatedby
ξ
and
η
[30,ParticularCase12.29].

REGULARITYOFOPTIMALTRANSPORT

3

Themainassumptionusedin[22,26,27]isthat
X(1.2)
S
(
x,y
)

K
|
ξ
|
2
|
η
|
2
whenever
c
i,j
ξ
i
η
j
=0
,
jiwhere
K
isapositiveconstant(strongMTWcondition)or
K
=0(weakMTW
condition).Condition(1.2)impliesthatthesectionalcurvatureof
M
isbounded
belowby
K
.Loeper[21]showedthattheroundsphere
S
n
satisfies(1.2)forsome
K>
0(seealso[32]).
Thereisbynowplentyofevidencethattheseconditions,complicatedastheyseem,
arenaturalassumptionstodeveloptheregularitytheoryofoptimaltransport.In
particular,Loeper[20]showedhowtoconstructcounterexamplestotheregularityif
theweakMTWconditionisnotsatisfied.Thefollowingprecisestatementisproven
in[30,Theorem12.39];volstandsfortheRiemannianvolumemeasure.
Theorem1.1
(Necessaryconditionfortheregularityofoptimaltransport)
.
Let
M
beaRiemannianmanifoldsuchthat
S
(
x,y
)

(
ξ,η
)
<
0
forsome
x,y,ξ,η
.Thenthere
are
C

positiveprobabilitydensities
f
and
g
on
M
suchthattheoptimaltransport
mapfrom

(
dx
)=
f
(
x
)vol(
dx
)
to
ν
(
dy
)=
g
(
y
)vol(
dy
)
,withcostfunction
c
=
d
2
,
isdiscontinuous.
(Forthesakeofpresentation,thistheoremisstatedin[30]underacompactness
assumption,buttheproofgoesthrougheasilytononcompactmanifolds.)
Conversely,smoothnessresultshavebeenobtainedundervarioussetsofassump-
tionsincludingeithertheweakorthestrongMTWcondition[10,16,19,20,22,27];
suchresultsarereviewedin[30,Chapter12].Forinstance,[22]furnishesinteriora
prioriregularityestimates(say
C
1
)ontheoptimaltransportmap,providedthatthe
optimaltransportplanissupportedinaset
D

M
×
M
suchthat(a)
c
isuniformly
smooth(say
C
4
)in
D
;(b)allsets(exp
x
)

1
(
D
x
)and(exp
y
)

1
(
D
y
)areconvex(in
T
x
M
and
T
y
M
respectively),where
D
x
=
{
y
;(
x,y
)

D
}
,
D
y
=
{
x
;(
x,y
)

D
}
,
andexpstandsfortheRiemannianexponential.(Themeaningofthenotation
(exp
x
)

1
willberecalledafterDefinition1.2.)Butsofar(a)and(b)havebeen
provenonlyinparticularcasessuchasthesphere
S
n
,oritsquotientslikethereal
projectivespace
RP
n
=
S
n
/

Id
}
[16,20].ThereisalsoapartialresultbyDelanoe¨
andGe[6]workingonperturbationsofthesphereandassumingcertainrestrictions
onthesizeofthedata.
Inthispaperwesuggestthata(possiblyslightlymodified)strictformoftheMTW
condition
alone
isanaturalsufficientconditionforregularity.Weshallprovethis
conjectureonlyunderasimplifyingnonfocalityassumptionwhichwenowexplain.
Tobeginwith,letusintroducesomenotation:

4

G.LOEPERANDC.VILLANI

Definition1.2
(injectivitydomain,tangentcutandfocalloci)
.
Let
M
beaRie-
mannianmanifoldand
x

M
.Forany
ξ

T
x
M
,
|
ξ
|
=1,let
t
C
(
ξ
)bethefirsttime
′t
suchthat(exp
x
(

))
0

s

t

isnotminimizingfor
t>t
;andlet
t
F
(
ξ
)

t
C
(
ξ
)be
thefirsttime
t
suchthat
d

exp
x
(thedifferentialofexp
x
at

)isnotone-to-one.
Wedefine
no
I(
x
)=

;0

t<t
C
(
ξ
)=injectivitydomainat
x
;
onTCL(
x
)=

;
t
=
t
C
(
ξ
)=

I(
x
)=tangentcutlocusat
x
;
onTFL(
x
)=

;
t
=
t
F
(
ξ
)=(first)tangentfocallocus
.
LetfurtherI(
x
)=I(
x
)

TCL(
x
).
ThenwedefineI(
M
)=

(
{
x

I(
x
)),TCL(
M
)=

(
{
x

TCL(
x
)),TFL(
M
)=

(
{
x

TFL(
x
)),andequipthesesetswiththetopologyinducedby
TM
.
Thedenominationof(tangent)injectivitydomainisjustifiedbythefactthatexp
x
isone-to-oneI(
x
)

M
\
cut(
x
).Wedenoteitsinverseby(exp
x
)

1
:
M
\
cut(
x
)

I(
x
).Explicitly,(exp
x
)

1
(
y
)istheuniquevelocity
v

T
x
M
suchthat(exp
x
tv
)
0

t

1
isminimizingandexp
x
v
=
y
.Byextension,if
y

cut(
x
),wedenoteby(exp
x
)

1
(
y
)
thesetofallvelocities
v
satisfyingthelatterproperties.Basicpropertiesofthe
injectivitydomainandtangentcutlocusarereviewedinAppendixC.
Definition1.3
(nonfocality)
.
Wesaythatthecutlocusof
M
is
nonfocal
(orjust
that
M
isnonfocal)ifTCL(
M
)

TFL(
M
)=

;orequivalentlyif
t
F
(
ξ
)
>t
C
(
ξ
)for
all(
x,ξ
)intheunittangentbundleof
M
.
Inthispaper,weprovethefollowingregularityresult:
Theorem1.4
(Sufficientconditionfortheregularityofoptimaltransport)
.
Let
M
beaRiemannianmanifoldsatisfyingthestrongMTWcondition,andwhosecutlocus
isnonfocal.Thenforanytwo
C

positiveprobabilitydensities
f
and
g
on
M
,the
optimaltransportmapfrom

(
dx
)=
f
(
x
)vol(
dx
)
to
ν
(
dy
)=
g
(
y
)vol(
dy
)
,withcost
function
c
=
d
2
,is
C

.
Beforegoingon,letuspausetoremarkthespectacularcontrastbetweenTheorem
1.1andTheorem1.4:dependingonjustthetuningoftheMa–Trudinger–Wang
condition,a“generic”solutionoftheoptimaltransportproblemwithsmoothdata
maybeeither
C

,ornotevencontinuous.
NowletuscommentontheassumptionsofTheorem1.4.Thenonfocalityassump-
tionmayseemridiculousatfirstsight,sinceitisneversatisfiedbycompactsimply

REGULARITYOFOPTIMALTRANSPORT

5

connectedmanifoldswithpositivecurvature,atleastinevendimension.(Thisresult
isduetoKlingenberg,withancestorsasoldasPoincare´;Weinstein[33,Section6]
collectsvarioussufficientconditionssothatthecutlocus
is
focal.)Thusouras-
sumptionsbasicallyneednontrivialtopology—somethingwhichisveryuncommon
inoptimaltransporttheory.Infact,thearchetypeofamanifoldsatisfyingthe
assumptionsofTheorem1.4istherealprojectivespace.
However,toadvocateforTheorem1.4,letuspointoutthat
(a)Asnotedin[6],itfollowsfromknownresultsinRiemanniangeometrythat
anycompactmanifoldwithnontrivialtopology,satisfyingastrongenough(positive)
curvaturepinchingassumption,hasnonfocalcutlocus.
(b)Theorem1.4isaparticularcaseofamoregeneralresult(Theorem1.8)which
coversallknown(non-flat)manifoldsforwhichthereisa
C

regularitytheoryof
optimaltransport.
(c)Theorem1.4isalsothefirstresultofitskindtoallowfor
perturbations
:if
M
satisfiestheassumptionsofTheorem1.4,thenany
C
4
perturbationof
M
willalso
satisfythem(forinstance,any
C
4
perturbationof
RP
n
).

Remark1.5.
Therehasbeenintenseactivitytofindexamplesofmanifoldssatisfy-
ingMTWconditions.Newexamplescanbefoundin[17],butatthetimeofwriting
theyarestillnotmany.Already,showingthatthespheresatisfiestheseconditions
wasnotatrivialproblem[20,32].
Remark1.6.
Inconnectionwithcomment(c)above,letusrecordthefollowing
openproblem:
IsthestrongMTWconditionstableunderperturbationsoftheRie-
mannianmetric?
WhatmakesthisquestionnontrivialisthefactthattheMTW
conditionisnonlocalandshouldholdarbitrarilyclosetothecutlocus,eventhough
thedependenceofthedistanceupontheRiemannianmetricmaybecomeverywild
asoneapproachesthecutlocus.Atanonfocalcutpointthisproblemisnotserious
(whichexplainscomment(c)),butatafocalpointthisbecomesnontrivial.In[6]
theMa–Trudinger–Wangtensoriscontrollednearthespherebytwoderivativesof
thesectionalcurvatures,ratherthanfourderivativesofthemetric;butthefocality
problemisleftunsolved.Usingacleverstrategy,FigalliandRifford[10]managed
toanswerourquestionpositivelywhen
M
=
S
2
.
1.2.
Cutlocus:mainresult.
TheproofofTheorem1.4isbasedonastriking
geometricpropertywhichhasinterestonitsown,andseemstobethefirstofits
:dnik

6

G.LOEPERANDC.VILLANI

Theorem1.7.
Let
M
beaRiemannianmanifoldwithnonfocalcutlocus,satisfy-
ingthestrongMa–Trudinger–Wangcondition.Thenthereis
κ>
0
suchthatall
injectivitydomains
I(
x
)
of
M
are
κ
-uniformlyconvex.
Toputthisresultinperspectivewithmorefamiliarresults,recallthatthestrong
MTWconditionisareinforcementoftheconditionofuniformlypositivesectional
curvature,whichimpliesanupperboundonthediameterofinjectivitydomains(this
isjustanawkwardwaytoreformulatetheBonnet–Myerstheorem).Tosummarize,
ifSectstandsforsectionalcurvature,
strongMTW=

Sect

κ>
0
⇓⇓uniformconvexityofI(
x
)=

boundondiameter
ApartfromthislinkwiththeBonnet–Myerstheorem,Theorem1.7issubstantially
differentfromallpreviouslyknownresultsorconjecturesinthefield:itdoesnot
bearonthesizeordimensionortopologicalstructureofthecutlocus,butonits
globalgeometricshape.Italsodisplaysa“positiveeffect”ofpositivecurvature;this
wassomewhatunexpected,sinceitisusuallynegativecurvaturewhichhasagood
impactonthestructureofthecutlocus(bypreventingfocalization).
ThistheoremwillbeproveninSection5.Thekeystepinourproofisakind
of“continuitymethod”setintheinjectivitydomains,wherethenormplaysthe
roleof“orderingparameter”,andthe
strict
convexityallowstokeeponincreasing
theparameter.AmoregeneralvariantofTheorem1.7,allowingforsomesort
offocalization(underassumptionswhichinparticularincludethesphere),willbe
proveninTheorem1.8.
1.3.
OutlineofproofofTheorem1.4.
Wenowexplaintheplanoftheproofof
Theorem1.4,andtheroleofTheorem1.7therein.Theproofisdividedintofive
steps,ofvariabledifficulty.
1.AccordingtoMcCann[24],theoptimaltransportmapbetween
f
(
x
)vol(
dx
)
and
g
(
y
)vol(
dy
)takestheform
(1.3)
T
(
x
)=exp
x
(

ψ
(
x
))
,
whereeachgeodesic(exp
x
(
t

ψ
(
x
)))
0

t

1
isminimizing,

standsforgradient,and
thesemiconvexfunction
ψ
solvesaweakformoftheMonge–Ampe`retypeequation
)4.1(
22

2

f
(
x
)
g
(exp

ψ
(
x
))
det

ψ
(
x
)+

xx
cx,
exp

ψ
(
x
)=

det

xy
cx,
exp

ψ
(
x
)

.

REGULARITYOFOPTIMALTRANSPORT

7

(Hereexp

ψ
(
x
)isashorthandforexp
x

ψ
(
x
),

2
standsforHessian,

x
2
forthe
Hessianwithrespecttothe
x
variable,etc.)
2.ThestrongMTWconditionimpliescertaininequalitiesbetweendistances(The-
orem3.1),andtheuniformconvexityofallinjectivitydomains(Theorem1.7).The
combinationofbothimpliesapropertyof
M
whichwecall
uniformregularity
(The-
orem4.4);itisanintrinsicandglobalreformulationofsimilarconditionsintroduced
earlierintheregularitytheoryofoptimaltransport.
3.Fromtheuniformregularityfollowsthecontinuityofoptimaltransport,and
infactthe
C
1
regularityof
ψ
(Theorem6.1).Thisstepisbasedonthestrategyof
Loeper[20],simplifiedbyKimandMcCann[16,Appendices],furthersimplifiedand
extendedinthepresentwork.
4.The
C
1
regularityof
ψ
(andtheassumptionon
M
)impliesthattheoptimal
transportstaysawayfromthecutlocus(Theorem7.1),soittakesplaceinadomain
where
c
is
C

,withuniformbounds.
5.Steps2and4makeitpossibletoapplythelocalaprioriestimatesofMa,
TrudingerandWang[22]in
C
k,β
(Ho¨lder)spaces,where
β

(0
,
1),and
k

N
isarbitrarilylarge.(Theseaprioriestimatesareestablishedforasmoothcost
functiondefinedinadomainof
R
n
×
R
n
;butbytheintrinsicnatureof
S
[16],[30,
Remark12.30]theyalsoapplytoacurvedgeometry.)Thenonemayconclude,using
argumentssimilartothosein[22],that
ψ
is
C

if
f
and
g
are.Thisconcludesthe
proof.
Inthesequel,weshallonlytreatSteps2to4oftheaboveoutlineofproof,since
thesearetheonlynovelsteps.This,togetherwiththeproofofTheorem1.7,will
occupySections3to7.
TheninSection8,weshallestablishthe
C
1

regularityof
ψ
withoutanysmooth-
nessassumptionontheprobabilitydensities,inthestyleof[20].
WeshallalsoestablishamoregeneralversionofTheorems1.4and1.7,whichhas
themerittocoveratthesametimethecaseofthesphere
S
n
.Letusdefine
1−(1.5)
δ
(
M
)=
(
x,v
)

i
T
n
C
f
L(
M
)
diam(exp
x
)(exp
x
v
)
.
Theorem1.8.
Let
M
beaRiemannianmanifoldsatisfyingcondition
MTW(
K
0
,C
0
)
ofSection2,forsome
K
0
,C
0
>
0
,suchthat
δ
(
M
)
definedin
(1.5)
ispositive,and
suchthatforany
x

M
,
TFL(
x
)
hasnonnegativesecondfundamentalformnear
TCL(
x
)

TFL(
x
)
.Then

8

G.LOEPERANDC.VILLANI

(a)thereis
κ>
0
suchthatallinjectivitydomainsof
M
are
κ
-uniformlyconvex;
(b)foranytwo
C

positiveprobabilitydensities
f
and
g
on
M
,theoptimal
transportmapfrom

(
dx
)=
f
(
x
)vol(
dx
)
to
ν
(
dy
)=
g
(
y
)vol(
dy
)
,withcostfunction
c
=
d
2
,is
C

.
HerearesomecommentsonTheorem1.8:

InthenotationofPropositionC.1,theassumption
δ
(
M
)
>
0standsbetween
J
=

(nofocalcutvelocity)and
J
\
Σ=

(nopurelyfocalcutvelocity).

PropositionC.5(a)andLemma2.3showthatTheorem1.8generalizesTheo-
rems1.4and1.7;inawaythisresultseemstobethebestthatonecanhopefor
withthetechniquesofthispaper.However,theassumptionsofTheorem1.8,unlike
thoseofTheorems1.4and1.7,arenotstableunderperturbation;thisisthereason
whywechosenottopresentitasourmainresult.

ThenonnegativityofthesecondfundamentalformofTFL(
x
)istobeunderstood
inweaksense(seetheremindersinAppendixA).Theextraassumptionputonthe
focalcutlocusisnotsobadasonemaythink,becausethefocallocusisamuchless
mysteriousobjectthanthecutlocus.
TheproofofTheorem1.8followsthesamegenerallinesastheproofsofTheorems
1.4and1.7,butdetailsaremuchmoretricky.Weshallsketchtheargumentsatthe
endofSections4,5and7.
TherearefourAppendices.Thefirsttwoaredevotedtovariousnotionsrelated
toconvexity.Inthethirdone,wegathersometechnicalresultsaboutthestructure
ofthetangentcutlocus.(Hopefullyourproblemswillconstituteamotivationto
pushthestudyofthistopic.)Inthefourthoneweconstructacounterexample
showingthatpositivesectionalcurvaturealonedoesnotguaranteetheconvexityof
injectivitydomains.
Acknowledgement:
Thisworkwasstartedduringastayofthesecondauthorin
Canberra(Summer2007),fundedbyaFASTresearchgrantcoordinatedbyPhilippe
Delanoe¨andNeilTrudinger.Thiswasagoldenopportunityforhimtolearnthe
subjectofregularityofoptimaltransportfromNeilTrudingerandXu-JiaWang.
WearegratefultoRobertMcCannandYoung-HeonKimforexchangingpreprints
andideas;specialthanksareduetoYoung-Heonforspottingaholeinaprelimi-
naryversionofthispaper.WewarmlythankLudovicRiffordandAlessioFigalli
foracarefulreadingandpreciouscommentsandcontributionsaboutthefocalcase;
wegladlynotethatourdiscussionsledtothegenesisof[10].Furtherenlightening