A Lack of Ricci Bounds for the Entropic Measure on Wasserstein space over the Interval
aa r X i v : . [ m a t h . M G ] M a r A LACK OF RICCI BOUNDS FOR THE ENTROPIC MEASURE ONWASSERSTEIN SPACE OVER THE INTERVAL
OTIS CHODOSH
Abstract.
We examine the entropic measure, recently constructed by von Renesseand Sturm, a measure over the metric space of probability measures on the unitinterval equipped with the 2-Wasserstein distance. We show that equipped withthis measure, Wasserstein space over the interval does not admit generalized Riccilower bounds in the entropic displacement convexity sense of Lott-Villani-Sturm. Wediscuss why this is contrary to what one might expect from heuristic considerations.
Contents
1. Introduction 11.1. Optimal Transport and Wasserstein Space 21.2. Metric Structure of ( P , d W ) 41.3. Entropic Measure 51.4. Heuristics 51.5. Acknowledgments 72. Proof of Theorem 1.1 73. Higher Dimensional Setting 9References 101. Introduction
For a compact Riemannian manifold M , the space of Borel probability measureson M , P ( M ), has very interesting geometric properties when equipped with what isknown as the 2-Wasserstein metric, arising via optimal transport theory. In particular,as discovered by Otto in [10], it can be formally regarded as an infinite dimensionalRiemannian manifold with well understood notions of tangent spaces, geodesics, gra-dients and more (in fact, many of these these formal notions have very recently beenmade rigorous in various settings, in addition to Otto’s original paper [10], also see[11, 7, 5] for discussions of both the formal structure and rigorous results along thesame lines). However, there is a notable lack of the notion of a volume measure in thisstructure. In fact, in [5, Remark 5.6], Gigli gives an argument that there should not bea natural volume measure on the space of probability measures, which we discuss in 1.4.If we cannot hope for a natural choice of volume measure “agreeing with the (formal) Date : October 21, 2018.2000
Mathematics Subject Classification.
Riemannian structure”, we can at least search for measures on P ( M ) with “nice” prop-erties. From a geometric point of view, one would be interested in a measure on P ( M )which would have interesting geometric properties and in particular, in this paper, wewill discuss the possibility of lower Ricci curvature bounds on P ( M ). The space P ( M )is only formally a Riemannian manifold, and it is therefore not a priori clear if thisis even a reasonable concept. However, thanks to the work of Lott-Villani-Sturm in[8, 12, 13], there is a well defined notion of a metric measure space having lower Riccibounds: convexity of the Boltzmann entropy functional along Wasserstein geodesicsin the space of probability measures on the space of interest. We will give definitionsbelow, but what will be most important about this concept in this paper is that itrequires a fixed background measure to make sense of it. Because there is no “Rie-mannian volume measure” on P ( M ), we should instead ask if there are choices whichhave interesting geometric properties, and in this paper we examine the possibility ofone such measure giving the space generalized lower Ricci bounds (which we will see itdoes not).In [18], von Renesse and Sturm have defined what they call an “entropic measure” P β on P := P ([0 , P β does not admit any lower Ricci bound. We will additionallydiscuss the possibility of extending our proof to the higher dimensional case for themeasure constructed by Sturm in [14] as a generalization of P β to arbitrary compactRiemannian manifolds M .Our main theorem will be (we will provide the necessary definitions in the remainderof the introduction) Theorem 1.1.
There is no K ∈ R , β > such that ( G , d L , Q β ) has generalized Ric ≥ K . Because we will see that this space is metric-measure isomorphic to ( P , d W , P β )(where the metric d W is the 2-Wasserstein distance, as defined below) we will have asan immediate corollary that: Corollary 1.2.
There is no K ∈ R , β > such that ( P , d W , P β ) has generalized Ric ≥ K . Optimal Transport and Wasserstein Space.
Below, we briefly review thebasic concepts of optimal transport, Wasserstein distance, and generalized Ricci boundsthrough displacement convexity of the entropy functional. For a proper introduction tothis topic we highly recommend that the reader refer to Villani’s book [15], as well ashis monograph [16]. In addition to these excellent sources, we remark that the materialcontained in this paper is a condensed form of the author’s essay, [1], which containsmore precise exposition and references for the introductory material below.Let (
X, d ) be a compact metric space. We will denote the space of probabilitymeasures on X by P ( X ). For k = 1 ,
2, we define proj k : X × X → X to be projectiononto the k -th factor. For two probability measures µ, ν ∈ P ( X ), we define the set of admissible transport plans to be(1.1) Π( µ, ν ) = { π ∈ P ( X × X ) : (proj ) ∗ π = µ, (proj ) ∗ π = ν } ACK OF RICCI BOUNDS FOR THE ENTROPIC MEASURE 3 where for k = 1 ,
2, (proj k ) ∗ π ∈ P ( X ) is the k -th marginal,(proj k ) ∗ ( A ) = π [(proj − k ( A )] . This is certainly nonempty, because the measure µ ⊗ ν which is defined by Z X × X f ( x, y ) dµ ⊗ ν ( x, y ) := Z X Z X f ( x, y ) dµ ( x ) dν ( y )for continuous f ∈ C ( X × X ), is clearly in Π( µ, ν ). Loosely speaking, a transport planis a proposal for how to move the mass of µ around so as to assemble the distribution ofmass prescribed by ν . Given the above definition, we define the 2- Wasserstein distance between µ and ν to be(1.2) d W ( µ, ν ) := inf π ∈ Π( µ,ν ) Z X × X d ( x, y ) dπ ( x, y ) . One can show that this defines a metric on P ( X ), metrizing the weak-* topology.Furthermore, if ( X, d ) is a geodesic space , meaning that for any two points x , x , thereis a continuous curve γ : [0 , → X with γ (0) = x , γ (1) = x and(1.3) d ( x , x ) = sup a = t X, d ) is a metricspace which is a geodesic space as defined above, and m ∈ P ( X ) is a fixed probabilitymeasure on X . We define the entropy functional Ent( ·| m ) : P ( X ) → R = R ∪ { + ∞} by µ (R X ρ log ρ dm for µ ≪ m , i.e. µ = ρm + ∞ otherwise.Intuitively, Ent( µ | m ) measures the nonuniformity of µ with respect to m . In some sense,we can think of it as a kind of distance (without symmetry or a triangle inequality),which is “extensive” in the sense that if µ, µ ′ , m ∈ P ( X ) then Ent( µ ⊗ µ ′ | m ⊗ m ) =Ent( µ | m ) + Ent( µ ′ | m ). Definition 1.3. For a measured geodesic space ( X, d, m ) , we say that Ent( ·| m ) isweakly a.c. K -displacement convex on ( X, d, m ) if for any probability measures µ , µ ≪ m , there exists a geodesic in P ( X ) , µ t from µ to µ so that Ent( µ t | m ) is K -convex,in the sense that (1.4) Ent( µ t | m ) ≤ t Ent( µ | m ) + (1 − t ) Ent( µ | m ) − K t (1 − t ) d W ( µ , µ ) . This definition is motivated by the amazing connection between displacement con-vexity of the entropy functional with lower Ricci bounds, when X is a Riemannianmanifold. The following theorem was first proven in this generality by von Renesse-Sturm in [17, Theorem 1], building work of Otto-Villani, [11], and Cordero-Erausquin-McCann-Schmuckenschl¨ager in [2]: OTIS CHODOSH Theorem 1.4. For a compact Riemannian manifold ( M, g ) , regarding it as a measuredgeodesic space ( M, d, m ) , with m = ] vol M ∈ P ( M ) , the normalized volume measure, we have that Ent( ·| m ) is weakly a.c. K -convex if and only if M has the lower Riccicurvature bound, Ric ≥ K on M . As such, we will often refer to a geodesic measure space ( X, d, µ ) for which Ent( ·| µ )is weakly a.c. K -convex as a “space with generalized Ric ≥ K ”.1.2. Metric Structure of ( P , d W ) . The principal reason that we work with measureson the unit interval is that the one dimensionality of the underlying space allows us usethe inverse distribution function to embed P := P ([0 , L [0 , Proposition 1.5. Letting G ⊂ L ([0 , be the subset of the square integrable functionson the interval which are right continuous and nondecreasing as maps g : [0 , → [0 , ,and let d L be the metric induced on G from the L norm, then the map Ψ : ( G , d L ) → ( P , d W )(1.5) g g ∗ Leb is an isometry. The inverse Ψ − is given by (1.6) Ψ − : µ g µ where g µ is the inverse distribution function defined (1.7) g µ ( s ) := inf { r ∈ [0 , 1] : µ ([0 , r ]) > s } with the convention that inf ∅ := 1 . For a proof, see [15, Theorem 2.18] or [1, Proposition 4.1]. It is not hard to showthat (see, for example [1, Lemma 4.2]): Lemma 1.6. The space G is a totally convex subset of L ([0 , (that is, any geodesicbetween two elements in G lies entirely in G ). In fact, for f, g ∈ G , the uniquegeodesic between them is given by the linear combination γ ( t ) := (1 − t ) f + tg. This fact proves highly beneficial to our analysis, and as we discuss in Section 3,its failure to hold in higher dimensions is one of the obstacles in extending our resultsto ( P ( M ) , d W , P β ), for M a general Riemannian manifold. As a result of the aboveproposition and lemma, instead of ( P , d W ), we can study the isometric space ( G , d L ),which is a convex subset of a Hilbert space, so has very simple geometry. That is, we define ] vol M := (vol( M )) − vol M . By this, we mean as a bilinear form, or in other words Ric ≥ K if and only if Ric( ξ, ξ ′ ) ≥ Kg ( ξ, ξ ′ )for all p ∈ M and ξ, ξ ′ ∈ T p M . ACK OF RICCI BOUNDS FOR THE ENTROPIC MEASURE 5 Entropic Measure. We now discuss the measure Q β ∈ P ( G ), originally con-structed by von Renesse and Sturm in [18, Proposition 3.4]. Definition-Proposition 1.7. For β > there is a (unique) probability measure Q β ∈P ( G ) which we will call the entropic measure (it could also be referred to as the lawof the Dirichlet process or as the Gibbs measure) such that for each partition of [0 , t < t < · · · < t N < t N +1 = 1 and for all bounded measurable functions u : [0 , N → R , we have that (1.8) Z G u ( g ( t ) , . . . , g ( t N )) d Q β ( g )= Γ( β ) Q Ni =0 Γ( β ( t i +1 − t i )) Z Σ N u ( x , . . . , x N ) N Y i =0 ( x i +1 − x i ) β ( t i +1 − t i ) − dx · · · dx N where we define Σ N := { ( x , . . . , x N ) ∈ [0 , N : 0 = x < x < · · · < x N < x N +1 = 1 } , and Γ( s ) = R ∞ t s − e − t dt is the Gamma function. In their paper, von Renesse and Sturm prove the above existence result using theKolmogorov extension theorem. We will denote the pushforward measure P β := Ψ ∗ Q β ,where Ψ is defined in Proposition 1.5. We are in reality interested in the metric measuretriple ( P , d W , P β ), but because of its linear structure as a convex subset of L , wewill find it far easier to work with the space ( G , d L , Q β ) (which is metric measureisomorphic to the space we are interested in, so provides an equivalent object for study).1.4. Heuristics. In [18], von Renesse and Sturm give a heuristic argument that P β isof the form(1.9) d P β ( µ ) = 1 Z β e − β Ent( µ | m ) d P ( µ )where Z β is a normalizing constant, and Ent( µ | m ) is the entropy of µ with respectto the Lebesgue measure and P is to be thought of as a “uniform measure” on P (which does not actually exist). At first sight, one should expect such a measure tobe displacement convex, because the Ent( µ | m ) is 0-convex on P by Theorem 1.4, andby [12, Proposition 4.14], multiplying by the exponential of a convex function will notdecrease generalized Ricci bounds, were they to exist for the hypothetical measure P .We also remark that the measure P β displays properties which are consistent withsuch lower bounds. Von Renesse and Sturm have constructed a symmetric Dirichletform in [18], ( E , D ( E )), given as the closure in L ( G ) of the quadratic form E ( F ) := Z G | D F ( g ) | L ([0 , d Q β ( g )with domain n F ( g ) = ϕ (cid:16) h f , g i L ([0 , , . . . , h f m , g i L ([0 , (cid:17) : m ≥ , ϕ ∈ C b ( R m ) , f k ∈ L o OTIS CHODOSH and where D F ( g ) is the L -Fr´echet derivative of F at g , which for F in the domaindescribed above is D F ( g )( x ) = m X i =1 ∂ i ϕ (cid:16) h f , g i L ([0 , , . . . , h f m , g i L ([0 , (cid:17) f i ( x ) . The existence of such a Dirichlet form is interesting for various reasons (e.g. see [4])but in our case, it is relevant because D¨oring and Stannat have shown that E satisfiesa Poincar´e inequality Theorem 1.8 ([3] Theorem 1.2) . The Dirichlet form constructed in [18] , E satisfies aPoincar´e inequality with constant less than β , i.e. for all F ∈ D ( E )Var Q β ( F ) ≤ β E ( F ) . as well as a log-Sobolev inequality Theorem 1.9 ([3] Theorem 1.4) . There exists a constant C (independent of β ) suchthat for F ∈ D ( E ) Z G F ( g ) log F ( g ) k F k L ( Q β ) d Q β ( g ) ≤ Cβ E ( F ) . Both of these theorems are properties that would hold true if ( G , d L , Q β ) hadgeneralized Ricci curvature bounds (c.f. [8, Corollary 6.12, Theorem 6.18]), (and inparticular, Theorem 1.8 would be the consequence of the space having generalizedRicci bounded below by β ). However, in spite of these heuristics, there are no such Ricci lower bounds, as we seein Theorem 1.1. We do remark that in [5], Gigli has argued that there is no naturalchoice of volume form on P ( M ), because this would be equivalent to there existing aLaplacian (by an integration by parts formula), which seems not to exist, because ofthe issues related to tracing a Hessian type object over an infinite dimensional space.In addition, he has written (in [5, Remark 5.5]) that Sturm has communicated to hima measure theoretic argument that the measure P β could not be the volume form on P (and the same for the higher dimensional analogue), and as such it seems we shouldstop searching for a volume measure on P ( M ). Of course, none of this precludes anarbitrary reference measures in P ( P ( M )) giving rise to lower Ricci bounds on P ( M )(in particular, a point mass, say δ Leb ∈ P ( P ([0 , We remark that the log-Sobolev and Poincar´e inequalities are certainly weaker conditions thangeneralized lower Ricci bounds. To see this, consider the possibility of such inequalities on a metricmeasure space ( X, d, m ). It is clear that for a measurable function on X , ρ ( x ) with 0 < α ≤ ρ ( x ) ≤ β < ∞ and R ρdm = 1, the space ( X, d, ρm ) admits log-Sobolev and Poincar´e inequalities if and onlyif ( X, d, m ) does (with constants changing in a manner easily prescribed by α, β . On the other hand,generalized Ricci bounds change (or are destroyed) in a much more sensitive manner depending on ρ ,and should be thought of a “higher order” condition than log-Sobolev/Poincar´e inequalities. We thankthe referee for drawing our attention to this point. ACK OF RICCI BOUNDS FOR THE ENTROPIC MEASURE 7 Acknowledgments. This paper is a consolidated version of my essay writtenfor Part III of the Mathematical Tripos at Cambridge University for the 2010-2011academic year. I would like to thank Cl´ement Mouhot for agreeing to set and markthe essay, assisting me in learning the material contained within, his extensive editinghelp, suggesting to me the problem of Ricci bounds on ( P ( X ) , d W ), as well as hiscontinued support after the conclusion of the program. I would additionally like tothank Cedric Villani, as well as the anonymous referee for their helpful comments andsuggestions. I am grateful to the Cambridge Gates Trust for their financial supportduring my year at Cambridge. Preparation of this paper was partially completed whilesupported by the National Science Foundation Graduate Research Fellowship underGrant No. DGE-1147470. 2. Proof of Theorem 1.1 Suppose Theorem 1.1 is false, so there is some β > , K ∈ R such that the abovespace has generalized Ric ≥ K . We will show that this yields a contradiction, as follows.Certainly, without loss of generality, we may assume that K ≤ 0. Let, for s ∈ (0 , A s := { g ∈ G : g ( s ) > / } B s := { g ∈ G : g ( s ) > } . It is clear that the following convex combination of these sets in G (or equivalently L by Lemma 1.6) is(2.1) (1 − t ) A s + tB s = { g ∈ G : g ( s ) > (1 − t ) / } := C s ( t ) . The significance of C s ( t ) is that any geodesic γ : [0 , → G such that γ (0) ∈ A s and γ (1) ∈ B s , has γ ( t ) ∈ C s ( t ). This will play a crucial role in our argument, allowing us toestimate the entropy of a Wasserstein geodesic in P ( X ) between Q β -uniform measuressupported on A s and B s .Notice that Q β ( C s ( t )) = Γ( β )Γ( βs )Γ( β (1 − s )) Z − t ) / x βs − (1 − x ) β (1 − s ) − dx. In particular, by Euler’s beta integral (see [6, Section 1.5]), we see that Q β ( B s ) = Q β ( C s (1)) = 1 , and for s ∈ (0 , 1) and it is not hard to see that for all t ∈ [0 , Q β ( C s ( t )) > 0. Assuch, we, define µ ( s ) := 1 Q β ( A s ) χ A s Q β ∈ P ( G )and by assumption, there is a geodesic µ ( s ) t between µ ( s ) and Q β such that the entropy,Ent( µ ( s ) t | Q β ) is K -convex, as in the definition of generalized Ric ≥ K . BecauseEnt( µ ( s ) | Q β ) = Z G Q β ( A s ) χ A s log Q β ( A s ) χ A s ! d Q β = − log( Q β ( A s )) < ∞ OTIS CHODOSH (and clearly Ent( Q β | Q β ) = 0) so by assumptions of K -convexity, we must have thatEnt( µ ( s ) t | Q β ) < ∞ , in particular implying that µ ( s ) t ≪ Q β . Thus, we can write µ ( s ) t = ρ ( s ) t Q β , and wehave that by (2.1), we see that µ ( s ) t is concentrated in C s ( t ) , implying thatEnt( µ ( s ) t | Q β ) = Z G ρ ( s ) t log ρ ( s ) t d Q β = Z G log ρ ( s ) t dµ ( s ) t = Z G log dµ ( s ) t d Q β | C s ( t ) Q β ( C s ( t )) ! dµ ( s ) t = Z G log dµ ( s ) t d Q β | C s ( t ) ! dµ ( s ) t − Z G log (cid:16) Q β ( C s ( t )) (cid:17) dµ ( s ) t = Ent( µ ( s ) t | Q β | C s ( t ) ) − log (cid:16) Q β ( C s ( t )) (cid:17) ≥ − log (cid:16) Q β ( C s ( t )) (cid:17) where Q β | C s ( t ) = Q β ( C s ( t )) χ C s ( t ) Q β . Combining this with the assumed K -convexity ofthe entropy functional along the path µ ( s ) t , we thus have that − log (cid:16) Q β ( C s ( t )) (cid:17) ≤ Ent( µ ( s ) t | Q β ) ≤ (1 − t ) Ent( µ ( s ) | Q β ) + t Ent( Q β | Q β ) | {z } =0 − K t (1 − t ) d ( µ ( s ) , Q β ) . This implies that, because Ent( µ ( s ) | Q β ) = − log (cid:16) Q β ( A s ) (cid:17) (2.2) log (cid:16) Q β ( C s ( t )) (cid:17) ≥ (1 − t ) log (cid:16) Q β ( A s ) (cid:17) + K t (1 − t ) d W ( µ ( s ) , Q β ) . Because diam( G ) = 1, we must have that d W ( µ ( s ) , Q β ) ∈ [0 , K ≤ Q β ( C s ( t ))( Q β ( A s )) − t ! ≥ K t (1 − t ) , This follows from the fact that optimal transport maps mass along geodesics of the underlyingspace. This is intuitively obvious, as if not, we could move along a geodesic between the endpoints,reducing the total distance traveled, and it follows rigorously from [8, Proposition 2.10]. Thus, µ ( s ) t is concentrated in ∪ γ γ ( t ) where the union is over all geodesics γ : [0 , → G with γ (0) ∈ A s and γ (1) ∈ B . Because G is a totally geodesic subset of a Hilbert space, we have that this union is just C s ( t ). ACK OF RICCI BOUNDS FOR THE ENTROPIC MEASURE 9 implying that for all s, t ∈ (0 , Q β ( C s ( t ))( Q β ( A s )) − t ≥ exp (cid:18) K t (1 − t ) (cid:19) . We will show that for a fixed t ∈ (0 , s → Q β ( C s ( t ))( Q β ( A s )) − t = 0 , contradicting (2.4), because the right hand side is bounded away from zero for a fixed t and K ∈ R . To see this, note that by definition of Q β , in Definition-Proposition 1.7,we have that(2.6) Q β ( C s ( t ))( Q β ( A s )) − t = (cid:18) Γ( β )Γ( βs )Γ( β (1 − s )) (cid:19) t R − t ) / x βs − (1 − x ) β (1 − s ) − dx (cid:16)R / x βs − (1 − x ) β (1 − s ) − (cid:17) − t . It is not hard to see that because we have fixed t ∈ (0 , t < x βs − term in the top integral becomes non-integrable at x = 0 as s → 0, but t < βs ) which approaches ∞ , showing (2.5), and thus showing that there cannot be any generalized Ricci lowerbounds on ( G , d L , Q β ). 3. Higher Dimensional Setting We remark that in [14], Sturm has constructed a higher dimensional analogue of P β ,over a general compact Riemannian manifold, M . We briefly describe his constructionand then explain why the method of proof of Theorem 1.1 to ( P ( M ) , d W , P β ) does notseem to extend to this case. In order to discuss Sturm’s construction, we first need thefollowing: Definition 3.1 ( d / . A function φ : M → R is called d / ifthere exists a function ψ : M → R so that φ ( y ) = inf x ∈ M (cid:20) d ( x, y ) − ψ ( x ) (cid:21) for all y ∈ M . For a function ψ : M → R , we define its d / φ d by thesame formula φ d ( y ) := inf x ∈ M (cid:20) d ( x, y ) − φ ( x ) (cid:21) . Theorem 3.2 ([9, Theorem 8]) . For µ, ν ∈ P ( M ) with µ ≪ vol M , there is a d / -concave function φ : M → R so that the map F t ( x ) := exp x ( − t ∇ φ ) gives µ t := ( F t ) ∗ µ : [0 , → P ( M ) , which is the unique geodesic between µ and ν .Furthermore, ( Id , F ) ∗ µ is an optimal transport plan between µ and µ . For t ∈ [0 , µ t ≪ vol M and if, in addition, ν ≪ vol M , then we have that for all t ∈ [0 , , µ t ≪ vol M . In fact, Sturm shows that ( P ( M ) , d W ) is homeomorphic to the space of maps ofthe form g = exp( −∇ φ ) for φ a d / M → R equipped with an H Sobolev norm (the homeomorphism is given by g g ∗ ] vol M , i.e. pushing forwardthe normalized volume measure, ] vol M := (vol M ( M )) − vol M , and the inverse is givenby finding the unique d / µ ∈ P ( M ) as given byTheorem 3.2). Sturm then notes that the d / P ( M ) → P ( M ), so for µ ∈ P ( M ) we map(3.1) C : µ exp( −∇ φ d ) ∗ ] vol M . Sturm then defines the measure Q β ∈ P ( P ( M )) by requiring that for each measurablepartition M = U Ni =1 M i and bounded Borel function u : R N → R , the following holds,where m i = ] vol M ( M i )(3.2) Z P ( M ) u ( ν ( M ) , . . . , ν ( M N )) d Q β ( ν ) =Γ( β ) Q Ni =1 Γ( βm i ) Z [0 , N , P Ni =1 x i =1 u ( x , . . . , x N ) x βm − · · · x βm N − N dx · · · dx N . Sturm argues that this measure exists by the Kolomogorov extension theorem (andEuler’s beta integral), and then defines the “multidimensional entropic measure” P β := C ∗ Q β .We conjecture that ( P ( M ) , d W , P β ) does not have an generalized lower Ricci bounds,but so far we have been unable to prove it. The principal difficulty seems to be that C ∗ : P ( P ( M )) → P ( P ( M )) does not seem to map Wasserstein geodesics to geodesics,so that if we were to try to mimic the one dimensional case, taking a geodesic from P β to a measure “close” to a singular measure, and then taking a sequence of suchgeodesics that end up closer and closer to the singular measure, to be able to computethe entropy we would have to push these geodesics forward by C ∗ , where they wouldno longer be geodesics. In particular, we remark that C ∗ makes the measure P β into avery nonlocal object, in the sense that it seems that there is no result implying thatif M ⊂ M is a totally convex subset and we were to know that ( P ( M ) , d W , P β ) hadgeneralized Ric ≥ K then we could conclude that ( P ( M ) , d W , P β | M ) (where P β | M isthe renormalization of P β restricted to the set of measures µ ∈ P ( M ) with µ ( M ) = 1).This is because the d / d / δ x is a singular measure which onecan show is supported on the cut locus of x and whose exact form seems to potentiallyhave a complicated relationship with the geometry of M ) which makes an argumentsimilar to our proof of Theorem 1.1 seem difficult in this higher dimensional setting. References [1] O. Chodosh , Optimal transport and Ricci curvature: Wasserstein space over the interval , http://arxiv.org/abs/1105.2883 , (2011).[2] D. Cordero-Erausquin, R. J. McCann, and M. 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Villani , Topics in optimal transportation , vol. 58 of Graduate Studies in Mathematics, Amer-ican Mathematical Society, Providence, RI, 2003.[16] , Optimal Transport: Old and New , vol. 338 of Grundlehren der Mathematischen Wis-senschaften [Fundamental Principles of Mathematical Sciences], Springer-Verlag, Berlin, 2009.[17] M.-K. von Renesse and K.-T. Sturm , Transport inequalities, gradient estimates, entropy, andRicci curvature , Comm. Pure Appl. Math, 58 (2005), pp. 923–940.[18] , Entropic measure and Wasserstein diffusion , Ann. Probab., 37 (2009), pp. 1114–1191. Department of Mathematics, Stanford University, CA 94305-2125, USA E-mail address ::