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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
))
dν
(
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
dν
=
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
,ν
i
)
}
i
∞
=1
beasequenceofcompactmeasuredlengthspaceswith
lim
i
→∞
(
X
i
,d
i
,ν
i
)=(
X,d,ν
)
inthemeasuredGromov–Hausdorfftopology.
(a)Forany
N
∈
[1
,
∞
)
,ifeach
(
X
i
,d
i
,ν
i
)
hasnonnegative
N
-Riccicurvaturethen
(
X,d,ν
)
hasnonnegative
N
-Riccicurvature.
(b)Ifeach
(
X
i
,d
i
,ν
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
dν
=
e
−
Ψ
dvol
B
forsomeΨ
∈
C
∞
(
B
):
Corollary0.14.
(a)If
(
B,g
B
,ν
)
isameasuredGromov–HausdorfflimitofRiemannian
manifoldswithnonnegativeRiccicurvatureanddimensionatmost
N
then
Ric
N
(
B
)
≥
0
.
(b)If
(
B,g
B
,ν
)
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
dν
=1
.
Theorem0.16.
Supposethatacompactmeasuredlengthspace
(
X,
R
d,ν
)
has
∞
-Riccicur-
(a)If
K>
0
then
2ZZ(0.17)
f
2
log(
f
2
)
dν
≤|∇
f
|
2
dν.
KXX(b)If
K
≤
0
then
1sZZ(0.18)
f
2
log(
f
2
)
dν
≤
2diam(
X
)
|∇
f
|
2
dν
−
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
dν
≤|∇
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
dγ
(
t
j
−
1
)
,γ
(
t
j
)
.
XJ
∈
N
0=
t
0
≤
t
1
≤
...
≤
t
J
=1
j
=1
Clearly
L
(
γ
)
≥
d
(
γ
(0)
,γ
(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
)
,γ
(
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
)
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