RICCI CURVATURE FOR METRIC MEASURE SPACES VIA OPTIMAL TRANSPORT
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RICCI CURVATURE FOR METRIC MEASURE SPACES VIA OPTIMAL TRANSPORT

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81 Pages
English

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Niveau: Supérieur, Doctorat, Bac+8
RICCI CURVATURE FOR METRIC-MEASURE SPACES VIA OPTIMAL TRANSPORT JOHN LOTT AND CEDRIC VILLANI Abstract. We define a notion of a measured length space X having nonnegative N -Ricci curvature, for N ? [1,∞), or having ∞-Ricci curvature bounded below by K, for K ? R. The definitions are in terms of the displacement convexity of certain functions on the associated Wasserstein metric space P2(X) of probability measures. We show that these properties are preserved under measured Gromov–Hausdorff limits. We give geometric and analytic consequences. This paper has dual goals. One goal is to extend results about optimal transport from the setting of smooth Riemannian manifolds to the setting of length spaces. A second goal is to use optimal transport to give a notion for a measured length space to have Ricci curvature bounded below. We refer to [11] and [44] for background material on length spaces and optimal transport, respectively. Further bibliographic notes on optimal transport are in Appendix F. In the present introduction we motivate the questions that we address and we state the main results. To start on the geometric side, there are various reasons to try to extend notions of curvature from smooth Riemannian manifolds to more general spaces. A fairly general setting is that of length spaces, meaning metric spaces (X, d) in which the distance between two points equals the infimum of the lengths of curves joining the points.

  • original metric

  • locally compact

  • length spaces

  • ricci curvature

  • riemannian manifold

  • transport

  • bounded below

  • optimal transport

  • compact measured

  • dimensional riemannian


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RICCICURVATUREFORMETRIC-MEASURESPACESVIAOPTIMAL
TRANSPORT

JOHNLOTTANDCE´DRICVILLANI

Abstract.
Wedefineanotionofameasuredlengthspace
X
havingnonnegative
N
-Ricci
curvature,for
N

[1
,

),orhaving

-Riccicurvatureboundedbelowby
K
,for
K

R
.
Thedefinitionsareintermsofthedisplacementconvexityofcertainfunctionsonthe
associatedWassersteinmetricspace
P
2
(
X
)ofprobabilitymeasures.Weshowthatthese
propertiesarepreservedundermeasuredGromov–Hausdorfflimits.Wegivegeometric
andanalyticconsequences.

Thispaperhasdualgoals.Onegoalistoextendresultsaboutoptimaltransportfromthe
settingofsmoothRiemannianmanifoldstothesettingoflengthspaces.Asecondgoalisto
useoptimaltransporttogiveanotionforameasuredlengthspacetohaveRiccicurvature
boundedbelow.Wereferto[11]and[44]forbackgroundmaterialonlengthspacesand
optimaltransport,respectively.Furtherbibliographicnotesonoptimaltransportarein
AppendixF.Inthepresentintroductionwemotivatethequestionsthatweaddressand
westatethemainresults.
Tostartonthegeometricside,therearevariousreasonstotrytoextendnotionsof
curvaturefromsmoothRiemannianmanifoldstomoregeneralspaces.Afairlygeneral
settingisthatof
lengthspaces
,meaningmetricspaces(
X,d
)inwhichthedistancebetween
twopointsequalstheinfimumofthelengthsofcurvesjoiningthepoints.Intherestof
thisintroductionweassumethat
X
isacompactlengthspace.Alexandrovgaveagood
notionofalengthspacehaving
“curvatureboundedbelowby
K

,with
K
arealnumber,
intermsofthegeodesictrianglesin
X
.InthecaseofaRiemannianmanifold
M
withthe
inducedlengthstructure,onerecoverstheRiemanniannotionofhavingsectionalcurvature
boundedbelowby
K
.LengthspaceswithAlexandrovcurvatureboundedbelowby
K
behavenicelywithrespecttotheGromov–Hausdorfftopologyoncompactmetricspaces
(moduloisometries);theyformaclosedsubset.
InviewofAlexandrov’swork,itisnaturaltoaskwhethertherearemetricspaceversions
ofothertypesofRiemanniancurvature,suchasRiccicurvature.Thisquestiontakes
substancefrom
Gromov’sprecompactnesstheorem
forRiemannianmanifoldswithRicci
curvatureboundedbelowby
K
,dimensionboundedaboveby
N
anddiameterbounded
aboveby
D
[23,Theorem5.3].Theprecompactnessindicatesthattherecouldbeanotion
ofalengthspacehaving“Riccicurvatureboundedbelowby
K
”,specialcasesofwhich
wouldbeGromov–HausdorfflimitsofmanifoldswithlowerRiccicurvaturebounds.
Date
:June23,2006.
TheresearchofthefirstauthorwassupportedbyNSFgrantDMS-0306242andtheClayMathematics
Institute.
1

2JOHNLOTTANDCE´DRICVILLANI

Gromov–HausdorfflimitsofmanifoldswithRiccicurvatureboundedbelowhavebeen
studiedbyvariousauthors,notablyCheegerandColding[15,16,17,18].Onefeatureof
theirwork,alongwiththeearlierworkofFukaya[21],isthatitturnsouttobeusefulto
addanauxiliaryBorelprobabilitymeasure
ν
andconsider
metric-measurespaces
(
X,d,ν
).
(AcompactRiemannianmanifold
M
hasacanonicalmeasure
ν
givenbythenormalized
Riemanniandensity
vdovl(ol
M
M
)
.)ThereisameasuredGromov–Hausdorfftopologyonsuch
triples(
X,d,ν
)(moduloisometries)andoneagainhasprecompactnessforRiemannian
manifoldswithRiccicurvatureboundedbelowby
K
,dimensionboundedaboveby
N
and
diameterboundedaboveby
D
.Hencethequestioniswhetherthereisagoodnotionofa
measuredlengthspace(
X,d,ν
)having
“Riccicurvatureboundedbelowby
K

.Whatever
definitiononetakes,onewouldlikethesetofsuchtriplestobeclosedinthemeasured
Gromov–Hausdorfftopology.Onewouldalsoliketoderivesomenontrivialconsequences
fromthedefinition,andofcourseinthecaseofRiemannianmanifoldsonewouldliketo
recoverclassicalnotions.Wereferto[16,Appendix2]forfurtherdiscussionoftheproblem
ofgivinga“synthetic”treatmentofRiccicurvature.
Ourapproachisintermsofametricspace(
P
(
X
)
,W
2
)thatiscanonicallyassociatedto
theoriginalmetricspace(
X,d
).Here
P
(
X
)isthespaceofBorelprobabilitymeasureson
X
and
W
2
istheso-called
Wassersteindistance
oforder2.ThesquareoftheWasserstein
distance
W
2
(

0
,
1
)between

0
,
1

P
(
X
)isdefinedtobetheinfimalcosttotransport
thetotalmassfromthemeasure

0
tothemeasure

1
,wherethecosttotransportaunit
ofmassbetweenpoints
x
0
,x
1

X
istakentobe
d
(
x
0
,x
1
)
2
.Atransportationscheme
withinfimalcostiscalledan
optimaltransport
.Thetopologyon
P
(
X
)comingfromthe
metric
W
2
turnsouttobetheweak-

topology.Wewillwrite
P
2
(
X
)forthemetricspace
(
P
(
X
)
,W
2
),whichwecallthe
Wassersteinspace
.If(
X,d
)isalengthspacethen
P
2
(
X
)
turnsouttoalsobealengthspace.Itsgeodesicswillbecalled
Wassersteingeodesics
.If
M
isaRiemannianmanifoldthenwewrite
P
2
ac
(
M
)fortheelementsof
P
2
(
M
)thatare
absolutelycontinuouswithrespecttotheRiemanniandensity.
Inthepastfifteenyears,optimaltransportofmeasureshasbeenextensivelystudiedin
thecase
X
=
R
n
,withmotivationcomingfromthestudyofcertainpartialdifferential
equations.Anotionwhichhasprovedusefulisthatof
displacementconvexity
,i.e.convex-
ityalongWassersteingeodesics,whichwasintroducedbyMcCanninordertoshowthe
existenceanduniquenessofminimizersforcertainrelevantfunctionson
P
2
ac
(
R
n
)[31].
Inthepastfewyears,someregularityresultsforoptimaltransporton
R
n
havebeen
extendedtoRiemannianmanifolds[19,32].Thismadeitpossibletostudydisplacement
convexityinaRiemanniansetting.OttoandVillani[36]carriedoutHessiancomputations
forcertainfunctionson
P
2
(
M
)usingaformalinfinite-dimensionalRiemannianstructure
on
P
2
(
M
)definedbyOtto[35].Theseformalcomputationsindicatedarelationshipbe-
tweentheHessianofan“entropy”functionon
P
2
(
M
)andtheRiccicurvatureof
M
.Later,
arigorousdisplacementconvexityresultforaclassoffunctionson
P
2
ac
(
M
),when
M
has
nonnegativeRiccicurvature,wasprovenbyCordero-Erausquin,McCannandSchmucken-
schla¨ger[19].ThisworkwasextendedbyvonRenesseandSturm[40].

RICCICURVATUREVIAOPTIMALTRANSPORT3
AgaininthecaseofRiemannianmanifolds,afurthercircleofideasrelatesdisplacement
convexitytologSobolevinequalities,Poincare´inequalities,Talagrandinequalitiesand
concentrationofmeasure[8,9,27,36].
Inthispaperweuseoptimaltransportanddisplacementconvexityinorderto
define
a
notionofameasuredlengthspace(
X,d,ν
)havingRiccicurvatureboundedbelow.If
N
is
afiniteparameter(playingtheroleofadimension)thenwewilldefineanotionof(
X,d,ν
)
havingnonnegative
N
-Riccicurvature.Wewillalsodefineanotionof(
X,d,ν
)having

-Riccicurvatureboundedbelowby
K

R
.(Theneedtoinputthepossibly-infinite
parameter
N
canbeseenfromtheBishop–Gromovinequalityforcomplete
n
-dimensional
RiemannianmanifoldswithnonnegativeRiccicurvature.Itstatesthat
r

n
vol(
B
r
(
m
))is
nonincreasingin
r
,where
B
r
(
m
)isthe
r
-ballcenteredat
m
[23,Lemma5.3.bis].When
wegofrommanifoldstolengthspacesthereisno
apriori
valuefortheparameter
n
.This
indicatestheneedtospecifyadimensionparameterinthedefinitionofRiccicurvature
bounds.)
Wenowgivethemainresultsofthepaper,sometimesinasimplifiedform.Forconsis-
tency,weassumeinthebodyofthepaperthattherelevantlengthspace
X
iscompact.
Thenecessarymodificationstodealwithcompletepointedlocallycompactlengthspaces
aregiveninAppendixE.
Let
U
:[0
,

)

R
beacontinuousconvexfunctionwith
U
(0)=0.Givenareference
probabilitymeasure
ν

P
(
X
),definethefunction
U
ν
:
P
2
(
X
)

R
∪{∞}
by
Z(0.1)
U
ν
(

)=
U
(
ρ
(
x
))

(
x
)+
U

(

)

s
(
X
)
,
Xerehw(0.2)

=
ρν
+

s
istheLebesguedecompositionof

withrespectto
ν
intoanabsolutelycontinuouspart
ρν
andasingularpart

s
,and
(0.3)
U

(

)=lim
U
(
r
)
.
r∞→rIf
N

[1
,

)thenwedefine
DC
N
tobethesetofsuchfunctions
U
sothatthefunction
(0.4)
ψ
(
λ
)=
λ
N
U
(
λ

N
)
isconvexon(0
,

).Wefurtherdefine
DC

tobethesetofsuchfunctions
U
sothatthe
function
(0.5)
ψ
(
λ
)=
e
λ
U
(
e

λ
)
isconvexon(
−∞
,

).Arelevantexampleofanelementof
DC
N
isgivenby
Nr
(1

r

1
/N
)if1
<N<

,
((0.6)
U
N
(
r
)=
r
log
r
if
N
=

.

4JOHNLOTTANDCE´DRICVILLANI

Definition0.7.
Given
N

[1
,

]
,wesaythatacompactmeasuredlengthspace
(
X,d,ν
)
hasnonnegative
N
-Riccicurvatureifforall

0
,
1

P
2
(
X
)
with
supp(

0
)

supp(
ν
)
and
supp(

1
)

supp(
ν
)
,thereis
some
Wassersteingeodesic
{

t
}
t

[0
,
1]
from

0
to

1
sothat
forall
U
∈DC
N
andall
t

[0
,
1]
,
(0.8)
U
ν
(

t
)

tU
ν
(

1
)+(1

t
)
U
ν
(

0
)
.
Given
K

R
,wesaythat
(
X,d,ν
)
has

-Riccicurvatureboundedbelowby
K
iffor
all

0
,
1

P
2
(
X
)
with
supp(

0
)

supp(
ν
)
and
supp(

1
)

supp(
ν
)
,thereis
some
Wassersteingeodesic
{

t
}
t

[0
,
1]
from

0
to

1
sothatforall
U
∈DC

andall
t

[0
,
1]
,
1(0.9)
U
ν
(

t
)

tU
ν
(

1
)+(1

t
)
U
ν
(

0
)

λ
(
U
)
t
(1

t
)
W
2
(

0
,
1
)
2
,
2where
λ
:
DC


R
∪{−∞}
isdefinedin
(5.14)
below.
Notethattheinequalities(0.8)and(0.9)areonlyassumedtoholdalong
some
Wasser-
steingeodesicfrom

0
to

1
,andnotnecessarilyalongallsuchgeodesics.Thisiswhatwe
call
weakdisplacementconvexity
.
Naturally,onewantstoknowthatinthecaseofaRiemannianmanifold,ourdefinitions
areequivalenttoclassicalones.Let
M
beasmoothcompactconnected
n
-dimensional
GivenΨ

C

(
M
)with
M
e

Ψ
dvol
M
=1,put

=
e

Ψ
dvol
M
.
manifoldwithRiemannia
R
nmetric
g
.Welet(
M,g
)denotethecorrespondingmetricspace.
Definition0.10.
For
N

[1
,

]
,letthe
N
-Riccitensor
Ric
N
of
(
M,g,ν
)
bedefinedby
1
Ric+Hess(Ψ)
if
N
=

,
(0.11)Ric=Ric+Hess(Ψ)

N

n
d
Ψ

d
Ψ
if
n<N<

,
N
Ric+Hess(Ψ)
−∞
(
d
Ψ

d
Ψ)
if
N
=
n,

−∞
if
N<n
,
wherebyconvention

0=0
.
Theorem0.12.
(a)For
N

[1
,

)
,themeasuredlengthspace
(
M,g,ν
)
hasnonnegative
N
-Riccicurvatureifandonlyif
Ric
N

0
.
(b)
(
M,g,ν
)
has

-Riccicurvatureboundedbelowby
K
ifandonlyif
Ric


Kg
.
InthespecialcasewhenΨisconstant,andso
ν
=
vdovl(ol
M
M
)
,Theorem0.12showsthat
werecovertheusualnotionofaRiccicurvatureboundfromourlengthspacedefinitionas
soonas
N

n
.
Thenexttheorem,whichisthemainresultofthepaper,saysthatournotionof
N
-Ricci
curvaturehasgoodbehaviorundermeasuredGromov–Hausdorfflimits.
Theorem0.13.
Let
{
(
X
i
,d
i

i
)
}
i

=1
beasequenceofcompactmeasuredlengthspaceswith
lim
i
→∞
(
X
i
,d
i

i
)=(
X,d,ν
)
inthemeasuredGromov–Hausdorfftopology.
(a)Forany
N

[1
,

)
,ifeach
(
X
i
,d
i

i
)
hasnonnegative
N
-Riccicurvaturethen
(
X,d,ν
)
hasnonnegative
N
-Riccicurvature.
(b)Ifeach
(
X
i
,d
i

i
)
has

-Riccicurvatureboundedbelowby
K
then
(
X,d,ν
)
has

-Ricci
curvatureboundedbelowby
K
.

RICCICURVATUREVIAOPTIMALTRANSPORT5
lovdTheorems0.12a

nd0.13im

plythatmeasuredGromov–Hausdorfflimits(
X,d,ν
)of
smoothmanifolds
M,g,
vol(
M
M
)
withlowerRiccicurvatureboundsfallunderourconsid-
erations.Additionally,weobtainthefollowingnewcharacterizationofsuchlimits(
X,d,ν
)
whichhappentobe
smooth
,meaningthat(
X,d
)isasmooth
n
-dimensionalRiemannian
manifold(
B,g
B
)and

=
e

Ψ
dvol
B
forsomeΨ

C

(
B
):
Corollary0.14.
(a)If
(
B,g
B

)
isameasuredGromov–HausdorfflimitofRiemannian
manifoldswithnonnegativeRiccicurvatureanddimensionatmost
N
then
Ric
N
(
B
)

0
.
(b)If
(
B,g
B

)
isameasuredGromov–HausdorfflimitofRiemannianmanifoldswithRicci
curvatureboundedbelowby
K

R
then
Ric

(
B
)

Kg
B
.
ThereisapartialconversetoCorollary0.14(seeCorollary7.45(ii,ii’)).
Finally,ifameasuredlengthspacehaslowerRiccicurvatureboundsthenthereare
analyticconsequences,suchasa
logSobolevinequality
.Tostateit,wedefinethe
gradient
norm
ofaLipschitzfunction
f
on
X
bytheformula
|
f
(
y
)

f
(
x
)
|
(0.15)
|∇
f
|
(
x
)=li
y
m

s
x
up
d
(
x,y
)
.
vatureboundedbelowby
K

R
.Supposethat
f

Lip(
X
)
satisfies
X
f
2

=1
.
Theorem0.16.
Supposethatacompactmeasuredlengthspace
(
X,
R
d,ν
)
has

-Riccicur-
(a)If
K>
0
then
2ZZ(0.17)
f
2
log(
f
2
)

≤|∇
f
|
2
dν.
KXX(b)If
K

0
then
1sZZ(0.18)
f
2
log(
f
2
)


2diam(
X
)
|∇
f
|
2


K
diam(
X
)
2
.
2XXInthecaseofRiemannianmanifolds,onerecoversfrom(0.17)thelogSobolevinequality
ofBakryandE´mery[6].
Aconsequenceof(0.17)isaPoincare´inequality.
curvatureboundedbelowby
K>
0
.Thenforall
h

Lip(
X
)
with
X
hdν
=0
,wehave
Corollary0.19.
Supposethatacompactmeasuredlengthspace
R
(
X,d,ν
)
has

-Ricci
1ZZ(0.20)
h
2

≤|∇
h
|
2
dν.
KXXInthecaseofRiemannianmanifolds,Corollary0.19followsfromtheLichnerowiczin-
equalityforthesmallestpositiveeigenvalueoftheLaplacian[28].
Wenowgivethestructureofthepaper.Moredetaileddescriptionsappearatthe
beginningsofthesections.
Section1givesbasicdefinitionsaboutlengthspacesandoptimaltransport.Section2
showsthattheWassersteinspaceofalengthspaceisalsoalengthspace,andthatWasser-
steingeodesicsarisefromdisplacementinterpolations.Section3definesweakdisplacement

6JOHNLOTTANDCE´DRICVILLANI

convexityanditsvariations.ThisisusedtoprovefunctionalinequalitiescalledtheHWI
inequalities.
Section4proves,modulothetechnicalresultsofAppendicesBandC,thatweakdisplace-
mentconvexityispreservedbymeasuredGromov–Hausdorfflimits.Thenotionof
N
-Ricci
curvatureisdefinedinSection5,whichcontainstheproofofTheorem0.13,alongwith
aBishop–Gromov-typeinequality.Section6proveslogSobolev,TalagrandandPoincare´
inequalitiesformeasuredlengthspaces,suchasTheorem0.16andCorollary0.19,along
withaweakBonnet–Myerstheorem.Section7looksatthecaseofsmoothRiemannian
manifoldsandproves,inparticular,Theorem0.12andCorollary0.14.
Therearesixappendicesthatcontaineithertechnicalresultsorauxiliaryresults.Ap-
pendixA,whichisasequeltoSection2,discussesthegeometryoftheWassersteinspace
ofaRiemannianmanifold
M
.Itshowsthatif
M
hasnonnegativesectionalcurvaturethen
P
2
(
M
)hasnonnegativeAlexandrovcurvature.Thetangentconesatabsolutelycontinuous
measuresarecomputed,therebymakingrigoroustheformalRiemannianmetricon
P
2
(
M
)
introducedbyOtto.
AppendicesBandCarethetechnicalcoreofTheorem0.13.AppendixBshowsthat
U
ν
(

)islowersemicontinuousinboth

and
ν
,andisnonincreasingunderpushforwardof

and
ν
.AppendixCshowsthatameasure


P
2
(
X
)withsupp(

)

supp(
ν
)canbe
weak-

approximatedbymeasures
{

k
}
k

=1
withcontinuousdensities(withrespectto
ν
)
sothat
U
ν
(

)=lim
k
→∞
U
ν
(

k
).
AppendixDcontainsformalcomputationsoftheHessianof
U
ν
.AppendixEexplains
howtoextendtheresultsofthepaperfromthesettingofcompactmeasuredlengthspaces
tothesettingofcompletepointedlocallycompactmeasuredlengthspaces.AppendixF
hassomebibliographicnotesonoptimaltransportanddisplacementconvexity.
Theresultsofthispaperwerepresentedattheworkshop“Collapsingandmetricge-
ometry”inMu¨nster,August1-7,2004.Afterthewritingofthepaperwasessentially
completedwelearnedofrelatedworkbyKarl-TheodorSturm[41,42].Also,Ludger
Ru¨schendorfkindlypointedouttousthatTheoremB.33wasalreadyprovenin[29,Chap-
ter1]bydifferentmeans.WedecidedtoretainourproofofTheoremB.33ratherthanjust
quoting[29],partlybecausethemethodofproofmaybeofindependentinterest,partly
forcompletenessandconveniencetothereader,andpartlybecauseourmethodofproof
isusedintheextensionofthetheoremconsideredinAppendixE.
WethankMSRIandtheUC-Berkeleymathematicsdepartmentfortheirhospitality
whilepartofthisresearchwasperformed.Wealsothanktheanonymousrefereesfortheir
suggestions.

Contents

1.Notationandbasicdefinitions
1.1.Convexanalysis
1.2.Geometryofmetricspaces
1.3.Optimaltransport:basicdefinitions
2.GeometryoftheWassersteinspace

8880111

47sdnuoberutavrucicciR-NfoytilibatS.5.E27stnemugranoitamixorppA.4.E17slanoitcnuF.3.E96ecapsnietsressaW.2.E86secapsdetnioP.1.E86esactcapmocnonehT.ExidneppA36snoitaluclacnaisseH.DxidneppA26seitisnedsuounitnocybnoitamixorppA.2.C16srefiilloM.1.C16)X(2PninoitamixorppA.CxidneppA06noitcartnocdnaytiunitnocimesrewoL.2.B65mrofsnarterdnegeLaivνUlanoitcnufehT.1.B65νUslanoitcnufehtfoseitreporpemoS.BxidneppA45)M(2PfonoitpircsedcirtemoegehtotnoitacilppA.3.A15sdlofinamdevrucylevitagennonfoesacehT.2.A05tropsnartlamitpodnasnoitcnufztihcspiL.1.A05ecapsvordnaxelAnasaecapsnietsressaWehT.AxidneppA74seiralloroccirtemoeG.2.714smretlacissalcnierutavrucicciR-NfonoitalumroF.1.704sdlofinamnainnameiRfoesacehT.704meroehtsreyM–tennoBkaeW.4.693∞<NesacehT.3.673∞=NesacehT.2.663seitilauqenilarenegehT.1.663seitilauqenie´racnioPdnadnargalaT,veloboSgoL.643ytiunitnocetulosbadnaytilibargetnimrofinU.6.513snoitcapuorgtcapmoC.5.592ytilauqenivomorG–pohsiB.4.592sdnuoberutavrucicciR-NfonoitavreserP.3.572ytixevnoctnemecalpsidkaewaiverutavrucicciR.2.562sessalcxevnoctnemecalpsiD.1.562secapshtgnelderusaemroferutavrucicciR-N.542ytixevnoctnemecalpsidkaewfoytilibatS.2.432ecapsnietsressaWehtfoecnegrevnocffrodsuaH–vomorG.1.422stimilffrodsuaH–vomorGderusaemdnaytixevnoctnemecalpsidkaeW.402seitilauqeniIWH.3.391selpmaxetnatropmI.2.351ytixevnoctnemecalpsidkaeW.1.351ecapsnietsressaWehtnoslanoitcnuF.351sdlofinamnainnameiRnotropsnartlamitpO.4.231snoitalopretnitnemecalpsidsascisedoegnietsressaW.3.221ecapshtgnelasaecapsnietsressaWehT.2.211snoitalopretnitnemecalpsiD.1.27TROPSNARTLAMITPOAIVERUTAVRUCICCIR
8JOHNLOTTANDCE´DRICVILLANI
E.6.TangentCones
AppendixF.Bibliographicnotesonoptimaltransport
References

777897

1.
Notationandbasicdefinitions
Inthissectionwefirstrecallsomefactsaboutconvexfunctions.Wethendefinegradient
norms,lengthspacesandmeasuredGromov–Hausdorffconvergence.Finally,wedefinethe
2-Wassersteinmetric
W
2
on
P
(
X
).
1.1.
Convexanalysis.
Letusrecallafewresultsfromconvexanalysis.See[44,Chapter
2.1]andreferencesthereinforfurtherinformation.
Givenaconvexlowersemicontinuousfunction
U
:
R

R
∪{∞}
(whichweassumeis
notidentically

),itsLegendretransform
U

:
R

R
∪{∞}
isdefinedby
(1.1)
U

(
p
)=sup
pr

U
(
r
)
.
R∈rThen
U

isalsoconvexandlowersemicontinuous.Wewillsometimesidentifyaconvex
lowersemicontinuousfunction
U
definedonaclosedinterval
I

R
withtheconvex
functiondefinedonthewholeof
R
byextending
U
by

outsideof
I
.
Let
U
:[0
,

)

R
beaconvexlowersemicontinuousfunction.Then
U
admitsa
leftderivative
U
′−
:(0
,

)

R
andarightderivative
U

+
:[0
,

)
→{−∞}∪
R
,with
U

+
(0
,

)

R
.Furthermore,
U
′−

U

+
.Theyagreealmosteverywhereandareboth
nondecreasing.Wewillwrite
′′
U
(
r
)
(1.2)
U
(

)=
r
li

m

U
+
(
r
)=
r
li

m

r

R
∪{∞}
.
Ifweextend
U
by


on(
−∞

,
0)thenitsLegendretransform
U

:
R

R
∪{∞}
becomes
U

(
p
)=sup
r

0
pr

U
(
r
).Itisnondecreasingin
p
,infiniteon(
U

(

)
,

)andequals

U
(0)on(
−∞
,U

+
(0)].Furthermore,itiscontinuouson(
−∞
,U

(

)).Forall
r

[0
,

),
wehave
U

(
U

+
(
r
))=
rU

+
(
r
)

U
(
r
).
1.2.
Geometryofmetricspaces.
1.2.1.
Gradientnorms.
Let(
X,d
)beacompactmetricspace(with
d
valuedin[0
,

)).
Theopenballofradius
r
around
x

X
willbedenotedby
B
r
(
x
)andthesphereofradius
r
around
x
willbedenotedby
S
r
(
x
).
Let
L

(
X
)denotethesetofboundedmeasurablefunctionson
X
.(Wewillconsider
suchafunctiontobedefinedeverywhere.)LetLip(
X
)denotethesetofLipschitzfunctions
on
X
.Given
f

Lip(
X
),wedefinethe
gradientnorm
of
f
by
(1.3)
|∇
f
|
(
x
)=limsup
|
f
(
y
)

f
(
x
)
|
y

x
d
(
x,y
)
if
x
isnotanisolatedpoint,and
|∇
f
|
(
x
)=0if
x
isisolated.Then
|∇
f
|∈
L

(
X
).

RICCICURVATUREVIAOPTIMALTRANSPORT9

Onsomeoccasionswewilluseafinernotionofgradientnorm:
(1.4)
|∇

f
|
(
x
)=limsup[
f
(
y
)

f
(
x
)]

=limsup[
f
(
x
)

f
(
y
)]
+
y

x
d
(
x,y
)
y

x
d
(
x,y
)
if
x
isnotisolated,and
|∇

f
|
(
x
)=0if
x
isisolated.Here
a
+
=max(
a,
0)and
a

=max(

a,
0).Clearly
|∇

f
|
(
x
)
≤|∇
f
|
(
x
).Notethat
|∇

f
|
(
x
)isautomatically
zeroif
f
hasalocalminimumat
x
.Inasense,
|∇

f
|
(
x
)measuresthedownwardpointing
componentof
f
near
x
.
1.2.2.
Lengthspaces.
If
γ
isacurvein
X
,i.e.acontinuousmap
γ
:[0
,
1]

X
,thenits
lengthis
J(1.5)
L
(
γ
)=supsup

(
t
j

1
)

(
t
j
)
.
XJ

N
0=
t
0

t
1

...

t
J
=1
j
=1
Clearly
L
(
γ
)

d
(
γ
(0)

(1)).
Wewillassumethat
X
isa
lengthspace
,meaningthatthedistancebetweentwopoints
x
0
,x
1

X
istheinfimumofthelengthsofcurvesfrom
x
0
to
x
1
.Suchaspaceispath
connected.
As
X
iscompact,itisastrictlyintrinsiclengthspace,meaningthatwecanreplace
infimumbyminimum[11,Theorem2.5.23].Thatis,forany
x
0
,x
1

X
,thereisaminimal
geodesic(possiblynonunique)from
x
0
to
x
1
.Wemaysometimeswrite“geodesic”instead
of“minimalgeodesic”.
By[11,Proposition2.5.9],anyminimalgeodesic
γ
joining
x
0
to
x
1
canbeparametrized
uniquelyby
t

[0
,
1]sothat
′′(1.6)
d
(
γ
(
t
)

(
t
))=
|
t

t
|
d
(
x
0
,x
1
)
.
Wewilloftenassumethatthegeodesichasbeensoparametrized.
Bydefinition,asubset
A

X
is
convex
ifforany
x
0
,x
1

A
thereisaminimizing
geodesicfrom
x
0
to
x
1
thatliesentirelyin
A
.Itis
totallyconvex
ifforany
x
0
,x
1

A
,any
minimizinggeodesicin
X
from
x
0
to
x
1
liesin
A
.
Given
λ

R
,afunction
F
:
X

R
∪{∞}
issaidtobe
λ
-convex
ifforanygeodesic
γ
:[0
,
1]

X
andany
t

[0
,
1],wehave
(1.7)
F
(
γ
(
t
))

tF
(
γ
(1))+(1

t
)
F
(
γ
(0))

1
λt
(1

t
)
L
(
γ
)
2
.
2Inthecasewhen
X
isasmoothRiemannianmanifoldwithRiemannianmetric
g
,and
F

C
2
(
X
),thisisthesameassayingthatHess
F

λg
.
1.2.3.
(Measured)Gromov–Hausdorffconvergence.
Definition1.8.
Giventwocompactmetricspaces
(
X
1
,d
1
)
and
(
X
2
,d
2
)
,an
ǫ
-Gromov–
Hausdorffapproximationfrom
X
1
to
X
2
isa(notnecessarilycontinuous)map
f
:
X
1

X
2
sothat
(i)Forall
x
1
,x

1

X
1
,

d
2
(
f
(
x
1
)
,f
(
x

1
))

d
1
(
x
1
,x

1
)