A Metric Stability Result for the Very Strict CD Condition
aa r X i v : . [ m a t h . M G ] J a n A Metric Stability Result for the Very Strict CDCondition
Mattia Magnabosco
Abstract
In [15] Schultz generalized the work of Rajala and Sturm [13], proving that a weak non-branching condition holds in the more general setting of very strict CD spaces. Anyway,similar to what happens for the strong CD condition, the very strict CD condition seems notto be stable with respect to the measured Gromov Hausdorff convergence (cf. [11]).In this article I prove a stability result for the very strict CD condition, assuming some metricrequirements on the converging sequence and on the limit space. The proof relies on the notionsof consistent geodesic flow and consistent plan selection , which allow to treat separately thestatic and the dynamic part of a Wasserstein geodesic. As an application, I prove that themetric measure space R N equipped with a crystalline norm and with the Lebesgue measuresatisfies the very strict CD (0 , ∞ ) condition. In their pivotal works Lott, Villani [10] and Sturm [18, 19] introduced a weak notion ofcurvature dimension bounds, which strongly relies on the theory of Optimal Transport. Theynoticed that, in a Riemannian manifold, a uniform bound on the Ricci tensor is equivalent tothe uniform convexity of the Boltzmann-Shannon entropy functional in the Wasserstein space.This allowed them to define a consistent notion of curvature dimension bound for metric measurespaces, that is known as CD condition. The metric measure spaces satisfying the CD conditionare called CD spaces and enjoy some remarkable analytic and geometric properties.The validity of the CD condition in a metric measure space ( X, d , m ) is strongly relatedto the metric structure of its Wasserstein space, which is in turn strictly dependent on themetric structure of ( X, d , m ). For this reason, it is not surprising that some properties of CDspaces hold under additional metric assumptions. Among them, one of the most important isthe non-branching condition, which basically prevents two different geodesic to coincide in aninterval of times. Since the first works on CD spaces, it has been clear that the non-branchingassumption, associated with the CD condition, could confer some nice properties to a metricmeasure space. For example, Sturm [18] was able to prove the tensorization property and thelocal-to-global property, while Gigli [8] managed to solve the Monge problem. The relationbetween non-branching assumption and CD condition was made even more interesting by thework of Rajala and Sturm [13]. They proved that the strong CD condition implies a weak versionof the non-branching one, that they called essentially non-branching. The work of Rajala andSturm was then generalized to the wider context of very strict CD spaces by Schultz in [15] (seealso [16] and [17], where he investigates some properties of very strict CD spaces). In particular,he also underlined that every very strict CD space satisfies a weak non-branching condition, thatI will call weak essentially non-branching .One of the most important properties of CD spaces is their stability with respect to themeasured Gromov Hausdorff convergence. Unfortunately this rigidity result cannot hold for thestrong CD condition and, accordingly to [11], it also does not hold for the so called strict CDcondition, which is (a priori) weaker than the very strict one, but stronger than the weak one.In particular, as explained in [11], it is not possible to deduce in general any type non-branchingcondition for a measured Gromov Hausdorff limit space. This motivates to add some analyticor metric assumption on the converging spaces, in order to achieve non-branching at the limit.1n this direction the most remarkable result is provided by the theory of RCD spaces, pioneeredby Ambrosio Gigli and Savar´e in [4] and [5]. In fact these spaces are stable with respect to themeasured Gromov Hausdorff convergence and essentially non-branching. In this article I presenta stability result for very strict CD spaces, assuming metric requirements on the convergingsequence and on the limit space.In particular, the first section is dedicated to introduce the necessary preliminary notions,related both to the Optimal Transport theory and to CD conditions. In this sense, this sectionshould be understood as a list of prerequisites and not as a complete treatment of the basic theory.For a full and precise discussions about the Optimal Transport theory I refer the reader [1], [2],[20] and [21].In the second section I prove a purely metric stability result, which assume some strongrigidity requirements, but nevertheless can be applied to a fair variety of metric measure spaces.This result relies on the notions of consistent geodesic flow and consistent plan selection , which,as it will be clear in the following, allow me to treat separately the dynamic and the staticparts of Wasserstein geodesics. The proof of this result uses an approximation strategy, and it iscompletely different from the arguments used for the RCD spaces theory.The result of the second section can be applied to the metric measure space R N equippedwith a crystalline norm and with the Lebesgue measure, this is explained in the last section.In particular I will show how a secondary variational problem can provide a consistent planselection, associated to the Euclidean consistent geodesic flow. This will allow to conclude thatthese metric measure spaces are very strict CD spaces, and therefore they are weakly essentiallynon-branching. This first section is aimed to state all the preliminary results I will need in the following.
The original formulation of the Optimal Transport problem, due to Monge, dates back to theXVIII century, and it is the following: given two topological spaces
X, Y , two probability measures µ ∈ P ( X ), ν ∈ P ( Y ) and a non-negative Borel cost function c : X × Y → [0 , ∞ ], look for themaps T that minimize the following quantityinf (cid:26)Z X c ( x, T ( x )) d µ ( x ) : T : X → Y Borel, T µ = ν (cid:27) . (M)The minimizers of the Monge problem are called optimal transport maps and in general do notnecessarily exist. Therefore for the development of a general theory, it is necessary to introducea slight generalization, due to Kantorovich. Defining the set of transport plans from µ to ν :Γ( µ, ν ) := { π ∈ P ( X × Y ) : ( p X ) π = µ and ( p Y ) π = ν } , the Kantorovich’s formulation of the optimal transport problem asks to find minima and mini-mizers of C ( µ, ν ) := inf (cid:26)Z X × Y c ( x, y ) d π ( x, y ) : π ∈ Γ( µ, ν ) (cid:27) . (K)This new problem admits minimizers under weak assumptions, in fact the following theoremholds. Theorem 1.1 (Kantorovich) . Let X and Y be Polish spaces and c : X × Y → [0 , ∞ ] a lowersemicontinuous cost function, then the minimum in the Kantorovich’s formulation (K) is attained. µ, ν ). Notice that this set obviously depends on the costfunction c , anyway I will usually avoid to make this dependence explicit, since many times it willbe clear from the context. A transport plan π ∈ Γ( µ, ν ) is said to be induced by a map if thereexists a µ -measurable map T : X → Y such that π = (id , T ) µ . Notice that these transportplans are exactly the ones considered in the Monge’s minimization problem (M). Remark . Suppose that every minimizer of the Kantorovich problem between the measures µ, ν ∈ P ( X ) is induced by a map, and thus is a minimizer for the Monge problem. Then the Kan-torovich problem between µ and ν admit a unique minimizer, which is clearly induced by a map.In fact, given two distinct transport plans π = (id , T ) µ, π = (id , T ) µ ∈ OptPlans( µ, ν ),their combination π = π + π is itself an optimal plan and it is not induced by a map, contradictingthe assumption.A fundamental approach in facing the Optimal Transport problem is the one of c -duality, whichallows to prove some very interesting and useful results. Before stating them let me introducethe notions of c -cyclical monotonicity, c -conjugate function and c -concave function. Definition 1.3.
A set Γ ⊂ X × Y is said to be c -cyclically monotone if N X i =1 c (cid:0) x i , y σ ( i ) (cid:1) ≥ N X i =1 c ( x i , y i )for every N ≥
1, every permutation σ of { , . . . , N } and every ( x i , y i ) ∈ Γ for i = 1 , . . . , N . Definition 1.4.
Given a function φ : X → R ∪ {−∞} , define its c -conjugate function φ c as φ c ( y ) := inf x ∈ X { c ( x, y ) − φ ( x ) } . Analogously, given ψ : Y → R ∪ {−∞} , define its c -conjugate function ψ c as ψ c ( x ) := inf y ∈ Y { c ( x, y ) − ψ ( y ) } . Notice that, by definition, given a function φ : X → R ∪ {−∞} , φ ( x ) + φ c ( y ) ≤ c ( x, y ) for every( x, y ) ∈ X × Y . Definition 1.5.
A function φ : X → R ∪ {−∞} is said to be c -concave if it is the infimum of afamily of c -affine functions c ( · , y ) + α . Analogously, ψ : Y → R ∪ {−∞} is said to be c -concave ifit is the infimum of a family of c -affine functions c ( x, · ) + β .The first important result of the c -duality theory is the following, which summarize the strictrelation that there is between optimality and c -cyclical monotonicity. Proposition 1.6.
Let X and Y be Polish spaces and c : X × Y → [0 , ∞ ] a lower semicontinuouscost function. Then every optimal transport plan π ∈ OptPlans( µ, ν ) such that R c d π < ∞ isconcentrated in a c -cyclically monotone set. Moreover, if there exist two functions a ∈ L ( X, µ ) and b ∈ L ( Y, ν ) such that c ( x, y ) ≤ a ( x ) + b ( x ) for every ( x, y ) ∈ X × Y , a plan π ∈ Γ( µ, ν ) isoptimal only if it is concentrated on a c -cyclically monotone set. The c -duality theory also allows to state the following duality result. Proposition 1.7.
Let X and Y be Polish spaces and c : X × Y → [0 , ∞ ] a lower semicontinuouscost function. If there exist two functions a ∈ L ( X, µ ) and b ∈ L ( Y, ν ) such that c ( x, y ) ≤ ( x ) + b ( x ) for every ( x, y ) ∈ X × Y , then there exists a c -concave function φ : X → R ∪ {−∞} satisfying C ( µ, ν ) = Z φ d µ + Z φ c d ν. Such a function φ is called Kantorovich potential. Remark . Notice that, if the cost c is continuous, every c -concave function is upper semicon-tinuous, being the infimum of continuous functions. Therefore, according to Proposition 1.7, itis possible to find an upper semicontinuous Kantorovich potential φ and its c -conjugate function φ c will be itself upper semicontinuous. In this section I am going to consider the Optimal Transport problem in the special case in which X = Y , ( X, d ) is a Polish metric space and the cost function is equal to the distance squared,that is c ( x, y ) = d ( x, y ). In this context the Kantorovich’s minimization problem induces the socalled Wasserstein distance on the space P ( X ) of probabilities with finite second order moment.Let me now give the precise definitions. Definition 1.9.
Define the set P ( X ) := (cid:26) µ ∈ P ( X ) : Z d ( x, x ) d µ ( x ) < ∞ for one (and thus all) x ∈ X (cid:27) Definition 1.10 (Wasserstein distance) . Given two measures µ, ν ∈ P ( X ) define their Wasser-stein distance W ( µ, ν ) as W ( µ, ν ) := min (cid:26)Z d ( x, y ) d π ( x, y ) : π ∈ Γ( µ, ν ) (cid:27) . Proposition 1.11. W is actually a distance on P ( X ) and ( P ( X ) , W ) is a Polish metricspace. The convergence of measures in P ( X ) with respect to the Wasserstein distance can be easilycharacterized and this is very useful in many situation. Proposition 1.12.
Let ( µ n ) n ∈ N ⊂ P ( X ) be a sequence of measures and let µ ∈ P ( X ) , then µ n W −−→ µ if and only if µ n ⇀ µ and Z d ( x, x ) d µ n → Z d ( x, x ) d µ for every x ∈ X. In particular, if ( X, d ) is a compact metric space, the Wasserstein convergence is equivalent toweak convergence. Let me now deal with the geodesic structure of ( P ( X ) , W ), which, as the following statementshows, is heavily related to the one of the base space ( X, d ). This fact makes the Wassersteinspace very important, and allows to prove many remarkable facts. First of all, notice that everymeasure π ∈ P ( C ([0 , , X )) induces a curve [0 , ∋ t → µ t = ( e t ) π ∈ P ( X ), therefore in thefollowing I will consider measures in P ( C ([0 , , X )) in order to consider curves in the Wassersteinspace. Proposition 1.13. If ( X, d ) is a geodesic space then ( P ( X ) , W ) is geodesic as well. Moreprecisely, given two measures µ, ν ∈ P ( X ) , the measure π ∈ P ( C ([0 , , X )) is a constant speed assertein geodesic connecting µ and ν if and only if it is concentrated in Geo( X ) (that is thespace of constant speed geodesics in ( X, d ) ) and ( e , e ) π ∈ OptPlans( µ, ν ) . In this case it is saidthat π is an optimal geodesic plan between µ and ν and this will be denoted as π ∈ OptGeo( µ, ν ) . Remark . I will say that an optimal geodesic plan π ∈ OptGeo( µ, ν ) is induced by a mapif there exists a µ -measurable map Θ : X → Geo( X ), such that π = Θ µ . Following theargument explained in Remark 1.2, it is possible to conclude that, if any optimal geodesic plan π ∈ OptGeo( µ, ν ) between two given measures µ, ν ∈ P ( X ) is induced by a map, then thereexists a unique π ∈ OptGeo( µ, ν ) and it is obviously induced by a map.Let me now introduce the entropy functional that will be the main character in definingthe notion of weak curvature dimension bounds. As it will be soon clear, the most appropriateframework in which deal with the entropy functional, is the one of metric measure spaces.
Definition 1.15.
A metric measure space is a triple ( X, d , m ), where ( X, d ) is a Polish metricspace and m is a non-negative and non-null Borel measure on X , finite on bounded sets. Moreover,a quadruple ( X, d , m , p ) is called pointed metric measure space if ( X, d , m ) is a metric measurespace and p is a point in X . Remark . In this article I assume that every metric measure space I am going to considersatisfies the following estimate Z e − c · d ( x,x ) d m ( x ) < ∞ , (1)for some (and thus all) x ∈ X and a suitable constant c >
0. This is essentially a technicalassumption, but it is useful to ensure the lower semicontinuity of the entropy functional (seeProposition 1.19).Let me now properly define the entropy functional.
Definition 1.17.
In a metric measure space ( X, d , m ), given a measure ν ∈ M ( X ) define therelative entropy functional with respect to ν Ent ν : P ( X ) → R asEnt ν ( µ ) := (R ρ log ρ d ν if µ ≪ ν and µ = ρν + ∞ otherwise . The entropy functional relative to the reference measure m will be simply denoted by Ent. Remark . According to [3], condition (1) prevents the entropy functional Ent to take thevalue −∞ .The most important property of the entropy functional is its lower semicontinuity with respectto the different notions of convergence in spaces of probabilities. Some results in this directionare collected in this proposition. Proposition 1.19. If m ( X ) < ∞ the functional Ent is lower semicontinuous with respect to theweak topology of P ( X ) , while if m ( X ) = ∞ (but (1) holds) Ent is lower semicontinuous withrespect to the Wasserstein convergence.
I conclude this subsection introducing two more spaces of probabilities, that will play animportant role in the following.
Definition 1.20.
Introduce the space P ac ( X ) ⊆ P ( X ) of probabilities absolutely continuouswith respect to m , with finite second order moments. Define also the space P ∗ ( X ) ⊆ P ac ( X ) as P ∗ ( X ) := { µ ∈ P ( X ) : Ent( µ ) < ∞} . .3 Curvature Dimension Bounds In this section I introduce the notions of curvature dimension bound and CD space, presentingalso the results which are the starting point of this work. Let me begin with the definition ofweak and strong CD condition.
Definition 1.21.
A metric measure space ( X, d , m ) is said to satisfy the (weak) CD ( K, ∞ ) con-dition and to be a (weak) CD ( K, ∞ ) space, if for every µ , µ ∈ P ∗ ( X ) there exists a Wassersteingeodesic with constant speed ( µ t ) t ∈ [0 , ⊂ P ∗ ( X ) connecting them, along which the relativeentropy functional is K -convex, that isEnt( µ t ) ≤ (1 − t ) Ent( µ ) + t Ent( µ ) − t (1 − t ) K W ( µ , µ ) , for every t ∈ [0 , . Moreover ( X, d , m ) is said to satisfy a strong CD ( K, ∞ ) condition and to be a strong CD ( K, ∞ )space if, for every µ , µ ∈ P ∗ ( X ), the relative entropy functional is K -convex along everyWasserstein geodesic with constant speed connecting them.The following proposition due to Villani [21] ensures the validity of CD condition in some familiarmetric measure spaces and it will be fundamental in the last section. Proposition 1.22.
Let k·k be a norm on R n and let d be the associated distance, then the metricmeasure space ( R n , d , L n ) is a (weak) CD (0 , ∞ ) space. The next result states the stability of CD condition with respect to the (pointed) measuredGromov Hausdorff convergence. I am not interested in giving a precise the definition of this notionof convergence, because in this article I will only deal with a different and stronger convergencefor metric measure spaces. For a precise definition I refer the reader to [21], where also the nexttheorem is proven. Let me also point out that the measured Gromov Hausdorff convergence canbe in some sense metrized by the D distance, introduced by Sturm in [19]. Moreover in [9] Gigli,Mondino and Savar´e showed that some different notion of convergence for pointed metric measurespaces are equivalent to the pointed measured Gromov Hausdorff convergence. Theorem 1.23.
Let ( X k , d k , m k , p k ) k ∈ N be a sequence of locally compact pointed metric mea-sure spaces converging in the pointed measured Gromov Hausdorff sense to a locally compactpointed metric measure space ( X, d , m , p ) . Given K ∈ R , if each ( X k , d k , m k ) k ∈ N satisfies theweak curvature dimension condition CD ( K, ∞ ) , then also ( X, d , m ) satisfies CD ( K, ∞ ) . I am now going to present the Rajala-Sturm theorem, which is the starting point of this work.In order to do this I have to preliminary introduce the notion of essentially non-branching metricmeasure space.
Definition 1.24.
A metric measure space ( X, d , m ) is said to be essentially non-branching if forevery absolutely continuous measures µ , µ ∈ P ac ( X ), every optimal geodesic plan η connectingthem is concentrated on a non-branching set of geodesics. Theorem 1.25.
Every strong CD ( K, ∞ ) metric measure space is essentially non-branching. The work of Rajala and Sturm was then generalized by Schultz [15] and applied to the contextof very strict CD spaces.
Definition 1.26.
A metric measure space ( X, d , m ) is called a very strict CD ( K, ∞ ) space iffor every absolutely continuous measures µ , µ ∈ P ac ( X ) there exists an optimal geodesic plan η ∈ OptGeo( µ , µ ), so that the entropy functional Ent satisfies the K-convexity inequality along6restr t t ) ( f η ) for every t < t ∈ [0 , f : Geo( X ) → R + with R f d η = 1.This CD condition is intermediate between the weak and the strong one and it easy to realizethat it cannot imply the essentially non-branching property. Anyway, as pointed out by Schultz,it is possible to prove a weaker version of the non-branching condition. Definition 1.27 (Weak Essentially Non-Branching) . A metric measure space ( X, d , m ) is said tobe weakly essentially non-branching if for every absolutely continuous measures µ , µ ∈ P ( X ),there exists an optimal geodesic plan connecting them, that is concentrated on a non-branchingset of geodesics. Theorem 1.28.
Every very strict CD ( K, ∞ ) space is weakly essentially non-branching. Unfortunately, as the reader can easily notice, the strong CD condition is not stable withrespect to the measured Gromov Hausdorff convergence. Moreover, the results in [11] suggestthat it is not possible to prove a general stability result also for the very strict CD condition. Theseobservations motivate this article, where I assume some metric requirements on the convergingsequence and on the limit space, in order to prove the very strict CD condition for suitablemeasured Gromov Hausdorff limit spaces.
In this section I state and prove some results that allow to prove very strict condition, andthus weak essentially non-branching, for some special measured Gromov Hausdorff limit spaces.These results do not assume any analytic requirement and are purely metric, therefore they canbe applied to a wide variety of metric measure spaces. The way to prove non-branching at thelimit in this case is very different from the one used by Ambrosio, Gigli and Savar´e in [5] and itis actually more straightforward.First of all, let me introduce two notions which provide a nice strategy to prove the verystrict CD condition, they are called consistent geodesic flow and consistent plan selection. As itwill be clear in the proof of Theorem 2.4, these two concepts allow to decouple the static partfrom the dynamic one, taking full advantage of Proposition 1.13. This, associated with suitableassumption, makes easier to deal with restriction of optimal geodesic plans and thus to prove thevery strict CD condition.
Definition 2.1.
Let ( X, d ) be a metric space. A measurable map G : X × X → C ([0 , , X ) iscalled consistent geodesic flow if the following properties hold:1) for every x, y ∈ X , G ( x, y ) is a constant speed geodesic connecting x and y , that is G ( x, y ) ∈ Geo( X ) with G ( x, y )(0) = x and G ( x, y )(1) = y ,2) restr ts G ( x, y ) = G (cid:0) G ( x, y )( s ) , G ( x, y )( t ) (cid:1) for every s < t ∈ (0 ,
1) and every x, y ∈ X .A consistent geodesic flow G is said to be L -Lipschitz ifsup t ∈ [0 , d (cid:0) G ( x , y )( t ) , G ( x , y )( t ) (cid:1) ≤ L · (cid:0) d ( x , x ) + d ( y , y ) (cid:1) , i.e. if it is an L -Lipschitz map considered as G : ( X × X, d ) → (cid:0) Geo( X ) , k·k sup (cid:1) , where d = d ⊗ d . 7 efinition 2.2. Let ( X, d , m ) be a metric measure space and assume there exists a consistentgeodesic selection G for the metric space ( X, d ). A map Π : P ac ( X ) × P ac ( X ) → P ( X × X ) iscalled consistent plan selection, associated to the flow G if1) Π( µ, ν ) ∈ OptPlans( µ, ν ) for every µ, ν ∈ P ac ( X )2) For every µ, ν ∈ P ac ( X ), every pair of times s < t ∈ [0 ,
1] and every bounded Borel function f : X × X → R + with R f dΠ( µ, ν ) = 1, if( G s ) (cid:0) f · Π( µ, ν ) (cid:1) , ( G t ) (cid:0) f · Π( µ, ν ) (cid:1) ∈ P ac ( X ) , where G r denotes the map e r ◦ G for every r ∈ [0 , G s , G t ) (cid:0) f · Π( µ, ν ) (cid:1) = Π (cid:0) ( G s ) (cid:0) f · Π( µ, ν ) (cid:1) , ( G t ) (cid:0) f · Π( µ, ν ) (cid:1)(cid:1) . Before going on I present the following lemma, that provides a useful equivalent characterizationfor condition 2 in the last definition.
Lemma 2.3.
Condition 2 in Definition 2.2 is equivalent to the combination of the following tworequirements2.1) f · Π( µ, ν ) = Π (cid:0) ( p ) (cid:0) f · Π( µ, ν ) (cid:1) , ( p ) (cid:0) f · Π( µ, ν ) (cid:1)(cid:1) for every µ, ν ∈ P ac ( X ) and everybounded Borel function f : X × X → R + with R f dΠ( µ, ν ) = 1 .2.2) For every µ, ν ∈ P ac ( X ) and every s < t ∈ [0 , , if ( G s ) Π( µ, ν ) , ( G t ) Π( µ, ν ) ∈ P ac ( X ) , then it holds ( G s , G t ) Π( µ, ν ) = Π (cid:0) ( G s ) Π( µ, ν ) , ( G t ) Π( µ, ν ) (cid:1) . Proof.
First of all notice that, putting f ≡ s = 0 and t = 1. Therefore condition 2 implies the combination of 2.1 and 2.2.On the other hand, assume that both 2.1 and 2.2 hold, then for every s < t ∈ [0 , µ, ν ∈ P ac ( X ) and every bounded Borel function f : X × X → R + with R f dΠ( µ, ν ) = 1, if( G s ) (cid:0) f · Π( µ, ν ) (cid:1) , ( G t ) (cid:0) f · Π( µ, ν ) (cid:1) ∈ P ac ( X ) , it holds that( G s , G t ) ( f · π ) = ( G s , G t ) Π (cid:0) ( p ) ( f · π ) , ( p ) ( f · π ) (cid:1) = Π (cid:18) ( G s ) Π (cid:0) ( p ) ( f · π ) , ( p ) ( f · π ) (cid:1) , ( G t ) Π (cid:0) ( p ) ( f · π ) , ( p ) ( f · π ) (cid:1)(cid:19) = Π (cid:18) ( G s ) ( f · π ) , ( G t ) ( f · π ) (cid:19) , where I have denoted by π the plan Π( µ, ν ), in order to ease the notation. This last relation isexactly the requirement of condition 2. (cid:3) I have introduced everything I need to prove one of the crucial results of this section. It showshow the existence of a consistent geodesic flow and a consistent plan selection, satisfying suitableassumptions, ensures the validity of the very strict CD condition.8 heorem 2.4.
Given a metric measure space ( X, d , m ) , assume there exist a consistent geodesicflow G for ( X, d ) and a consistent plan selection Π associated to G . Suppose also that for everypair of measures µ, ν ∈ P ac ( X ) , the K -convexity inequality of the entropy is satisfied along theWasserstein geodesic G Π( µ, ν ) for a suitable K , that is Ent (cid:0) ( G t ) Π( µ, ν ) (cid:1) ≤ (1 − t ) Ent( µ ) + t Ent( ν ) − K t (1 − t ) W ( µ, ν ) , for every t ∈ (0 , . Then ( X, d , m ) is a very strict CD ( K, ∞ ) space. Proof.
Fix two measures µ, ν ∈ P ac ( X ) and call π = Π( µ, ν ). Then I need to prove that the K -convexity inequality of the entropy holds along the optimal geodesic plan (restr ts ) (cid:0) f · G π (cid:1) , forevery s < t ∈ [0 ,
1] and every bounded Borel function f : C ([0 , , X ) → R + with R f d π = 1. Thisis obviously true when at least one of its marginals at time 0 and 1 is not absolutely continuous,therefore I can assume that( e s ) (cid:0) f · G π (cid:1) , ( e t ) (cid:0) f · G π (cid:1) ∈ P ac ( X ) . (2)In particular this allows me to apply condition 2 in Definition 2.2. Now notice that, since G isobviously injective, if one calls ˜ f = f ◦ G it holds(restr ts ) (cid:0) f · G π (cid:1) = (restr ts ) (cid:0) G ( ˜ f · π ) (cid:1) = (restr ts ◦ G ) ( ˜ f · π ) . Observe now that the definition of consistent geodesic flow ensures that restr ts ◦ G = G ◦ ( G s , G t ),thus (restr ts ) (cid:0) f · G π (cid:1) = ( G ◦ ( G s , G t )) ( ˜ f · π ) = G (cid:0) ( G s , G t ) ( ˜ f · π ) (cid:1) = G Π (cid:0) ( G s ) ( ˜ f · π ) , ( G t ) ( ˜ f · π ) (cid:1) . On the other hand it is obvious that( G s ) ( ˜ f · π ) = ( e s ) (cid:0) G ( ˜ f · π ) (cid:1) = ( e s ) (cid:0) f · G π (cid:1) , and similarly ( G t ) ( ˜ f · π ) = ( e t ) (cid:0) f · G π (cid:1) , so I can conclude that(restr ts ) (cid:0) f · G π (cid:1) = G Π (cid:0) ( e s ) (cid:0) f · G π (cid:1) , ( e t ) (cid:0) f · G π (cid:1)(cid:1) . At this point the fact that the entropy functional is K -convex along (restr ts ) (cid:0) f · G π (cid:1) is an easyconsequence of the assumption of the theorem, associated with (2). (cid:3) In the remaining part of the section I show how Theorem 2.4 can be applied in order toprove the very strict CD condition for some suitable measured Gromov Hausdorff limit spaces.The first result I want to present provides sufficient conditions to ensure the existence of aconsistent geodesic flow for a limit space. The reader must notice that I am considering a notionof convergence, that is much stronger than the measured Gromov Hausdorff convergence. Thischoice allows me to avoid some tedious technical details, but it is easy to notice that this resultcan be somehow extended to measured Gromov Hausdorff limit spaces. Anyway, as the nextsection confirms, in many easy applications this stronger hypothesis is sufficient.
Proposition 2.5.
Let ( X, d , m ) be a compact metric measure space and let { d n } n ∈ N be a se-quence of distances on X (inducing the same topology) such that there exist a sequence { ε n } n ∈ N converging to zero satisfying | d n ( x, y ) − d ( x, y ) | < ε n for every x, y ∈ X, n particular the sequence ( X, d n , m ) measured Gromov Hausdorff converges to ( X, d , m ) by meansof the identity map. Given L > , assume that for every n there exists an L -Lipschitz consistentgeodesic flow G n for the metric measure space ( X, d n , m ) . Then there exists an L -Lipschitzconsistent geodesic flow G for the metric measure space ( X, d , m ) and, up to subsequences, G n converges uniformly to G . Proof.
The space X × X is compact and thus separable, therefore there exists a countable denseset D ⊆ X × X . Notice that for every ( x, y ) ∈ X × X and every t ∈ [0 , { G n ( x, y )( t ) } n ∈ N is contained in the compact set X . Then the diagonal argument ensures that,up to taking a suitable subsequence, there exists a function G : D → { f : [0 , ∩ Q → X } such that for every ( x, y ) ∈ D and every t ∈ [0 , ∩ Q it holds G n ( x, y )( t ) → G ( x, y )( t ) . Now for every ( x, y ) ∈ D the function G ( x, y ) is a d ( x, y )-Lipschitz function on [0 , ∩ Q , in factfor every s, t ∈ [0 , ∩ Q it holds d (cid:0) G ( x, y )( s ) , G ( x, y )( t ) (cid:1) ≤ d (cid:0) G n ( x, y )( s ) , G ( x, y )( s ) (cid:1) + d (cid:0) G n ( x, y )( s ) , G n ( x, y )( t ) (cid:1) + d (cid:0) G n ( x, y )( t ) , G ( x, y )( t ) (cid:1) ≤ d (cid:0) G n ( x, y )( s ) , G ( x, y )( s ) (cid:1) + d n (cid:0) G n ( x, y )( s ) , G n ( x, y )( t ) (cid:1) + ε n + d (cid:0) G n ( x, y )( t ) , G ( x, y )( t ) (cid:1) = d (cid:0) G n ( x, y )( s ) , G ( x, y )( s ) (cid:1) + d n ( x, y ) · | t − s | + ε n + d (cid:0) G n ( x, y )( t ) , G ( x, y )( t ) (cid:1) → d ( x, y ) · | t − s | . This allows to extend G ( x, y ) to a d ( x, y )-Lipschitz function on the whole interval [0 , G ( x, y )(0) = x and G ( x, y )(1), I can infer that G ( x, y ) ∈ Geo( X ). Then for every( x, y ) ∈ D it is possible to extend the pointwise convergence of G n ( x, y ) to G ( x, y ) to the interval[0 , ε >
0, for every t ∈ [0 ,
1] there exists s ∈ [0 , ∩ Q with | t − s | < ε d ( x,y ) thatallows to perform the following estimate d (cid:0) G n ( x, y )( t ) , G ( x, y )( t ) (cid:1) ≤ d (cid:0) G n ( x, y )( t ) , G n ( x, y )( s ) (cid:1) + d (cid:0) G n ( x, y )( s ) , G ( x, y )( s ) (cid:1) + d (cid:0) G ( x, y )( s ) , G ( x, y )( t ) (cid:1) ≤ ( d n ( x, y ) + d ( x, y )) · | t − s | + d (cid:0) G n ( x, y )( s ) , G ( x, y )( s ) (cid:1) → d ( x, y ) · | t − s | < ε, the claim follows from the arbitrariness of ε . I end up with the map G : ( D, d ) → (cid:0) Geo( X ) , k·k sup (cid:1) ⊂ (cid:0) C ([0 , , X ) , k·k sup (cid:1) that is the pointwise limit of the L -Lipschitz maps G n , thus also G is an L -Lipschitz map.Therefore it can be extended to an L -Lipschitz function on the whole space X × X , furthermore,since Geo( X ) is closed with respect to the sup norm, I obtain G : ( X × X, d ) → (cid:0) Geo( X ) , k·k sup (cid:1) . Then the equicontinuity of the maps G n ensures that the sequence { G n } n ∈ N uniformly convergesto G . 10n order to conclude the proof I only need to show that G is a consistent geodesic flow,proving property 3 of Definition 2.1. To this aim fix ( x, y ) ∈ X × X , s, t ∈ [0 ,
1] and a small ε >
0, subsequently take n ∈ N such that k G n − G k sup < ε . Then it holds that (cid:13)(cid:13) restr ts G ( x, y ) − G (cid:0) G ( x, y )( s ) , G ( x, y )( t ) (cid:1)(cid:13)(cid:13) ≤ k G n − G k sup + (cid:13)(cid:13) restr ts G n ( x, y ) − G n (cid:0) G ( x, y )( s ) , G ( x, y )( t ) (cid:1)(cid:13)(cid:13) = 2 k G n − G k sup + (cid:13)(cid:13) G n (cid:0) G n ( x, y )( s ) , G n ( x, y )( t ) (cid:1) − G n (cid:0) G ( x, y )( s ) , G ( x, y )( t ) (cid:1)(cid:13)(cid:13) ≤ (2 + 2 L ) · k G n − G k sup < (2 + 2 L ) ε, thesis follows from the arbitrariness of ε . (cid:3) Once Proposition 2.5 has provided a consistent geodesic flow for the limit space, the next resultshows how, under suitable assumptions, it is possible to prove the very strict CD condition forthe metric measure space ( X, d , m ). Proposition 2.6.
Under the same assumptions of the last proposition, suppose that there existsa consistent plan selection Π on ( X, d , m ) , associated to G , such that for every µ, ν ∈ P ac ( X ) there exists a sequence π n ∈ OptPlans d n ( µ, ν ) satisfying1. π n ⇀ Π( µ, ν ) (up to the extraction of a subsequence),2. the K -convexity of the entropy functional holds along the d n -Wasserstein geodesic ( G n ) π n ,with respect to the distance d n .Then the metric measure space ( X, d , m ) is a very strict CD ( K, ∞ ) space. Proof.
Fix a time t ∈ [0 ,
1] and notice that the assumption 2 ensures thatEnt (cid:0)(cid:2) ( G n ) t (cid:3) π n (cid:1) ≤ (1 − t ) Ent( µ ) + t Ent( ν ) − K t (1 − t )( W d n ) ( µ, ν ) . (3)Now, since in compact space weak convergence and Wasserstein convergence coincide, it holdsthat W ( π n , Π( µ, ν )) →
0. Then taking an optimal transport plan η between π n and Π( µ, ν ) andhaving in mind that G is L -Lipschitz, it is possible to do the following estimate W (cid:0) ( G t ) π n , ( G t ) Π( µ, ν ) (cid:1) ≤ Z d (cid:0) G t ( x , y ) , G t ( x , y ) (cid:1) d η (cid:0) ( x , y ) , ( x , y ) (cid:1) ≤ Z L · d (cid:0) ( x , y ) , ( x , y ) (cid:1) d η (cid:0) ( x , y ) , ( x , y ) (cid:1) = L · W ( π n , Π( µ, ν )) → . Consequently, I am able to infer that W (cid:0)(cid:2) ( G n ) t (cid:3) π n , ( G t ) Π( µ, ν ) (cid:1) ≤ W (cid:0)(cid:2) ( G n ) t (cid:3) π n , ( G t ) π n (cid:1) + 2 W (cid:0) ( G t ) π n , ( G t ) Π( µ, ν ) (cid:1) ≤ Z d (cid:0) ( G n ) t , G t (cid:1) d π n + 2 W (cid:0) ( G t ) π n , ( G t ) Π( µ, ν ) (cid:1) ≤ k G n − G k + +2 W (cid:0) ( G t ) π n , ( G t ) Π( µ, ν ) (cid:1) → , and thus that (cid:2) ( G n ) t (cid:3) π n W −−→ ( G t ) Π( µ, ν ). Finally, since obviously ( W d n ) ( µ, ν ) → W ( µ, ν ),it is possible to pass to the limit in (3) using the lower semicontinuity of the entropy and obtainEnt (cid:0) ( G t ) Π( µ, ν ) (cid:1) ≤ (1 − t ) Ent( µ ) + t Ent( ν ) − K t (1 − t ) W ( µ, ν ) , which, associated to Theorem 2.4, allows to conclude the proof because t is arbitrary. (cid:3) Corollary 2.7.
Under the same assumptions of Proposition 2.5, suppose that there exists aconsistent plan selection Π on ( X, d , m ) , associated to G . Moreover assume that for every µ, ν ∈ P ac ( X ) there exist three sequences µ n , ν n ∈ P ac ( X ) and π n ∈ OptPlans d n ( µ n , ν n ) satisfying1. µ n ⇀ µ , ν n ⇀ ν and Ent( µ n ) → Ent( µ ) , Ent( ν n ) → Ent( ν ) ,2. π n ⇀ Π( µ, ν ) (up to the extraction of a subsequence),3. the K -convexity of the entropy functional holds along the d n -Wasserstein geodesic ( G n ) π n .Then the metric measure space ( X, d , m ) is a very strict CD ( K, ∞ ) space. As already anticipated before, similar results can be proven for suitable measured GromovHausdorff limit spaces, also in the non-compact case. These generalizations require some technicalassumption but their proof basically follow the proofs I have just presented. Anyway, in order to beconcise, I prefer not to present the most general statements, except for the following proposition,which will be fundamental in the next section. The reader can easily notice that it can be provenfollowing the proof of Proposition 2.6, except for two technical details that I will fix below.
Proposition 2.8.
Let ( X, d , m ) be a locally compact metric measure space and let { d n } n ∈ N bea sequence of distances on X (inducing the same topology), locally uniformly convergent to d as n → ∞ , such that there exists a constant H satisfying d n ( x, y ) ≤ H d ( x, y ) for every x, y ∈ X × X, (4) for every n . Assume that there exists a map G : X × X → C ([0 , , X ) which is a Lipschitzconsistent geodesic flow for d and a consistent geodesic flow for every distance d n . Moreover,suppose that there is a consistent plan selection Π on ( X, d , m ) , associated to G , such that forevery µ, ν ∈ P ac ( X ) there exists a sequence π n ∈ OptPlans d n ( µ, ν ) , satisfying1. π n ⇀ Π( µ, ν ) (up to the extraction of a subsequence),2. the K -convexity of the entropy functional holds along the d n -Wasserstein geodesic G π n ,with respect to the distance d n .Then the metric measure space ( X, d , m ) is a very strict CD ( K, ∞ ) space. Remark . Notice that condition (4) ensures that P ac ( R N , d ) ⊆ P ac ( R N , d n ) for every n . Proof.
In order to repeat the same strategy used for Proposition 2.6 I only need to prove that W ( π n , Π( µ, ν )) → n →∞ ( W d n ) ( µ, ν ) = W ( µ, ν ). For the first condition, accordingto Proposition 1.12, it is sufficient to prove that Z d (cid:0) ( x, y ) , ( x , y ) (cid:1) d π n ( x, y ) → Z d (cid:0) ( x, y ) , ( x , y ) (cid:1) dΠ( µ, ν )( x, y ) , for every fixed ( x , y ) ∈ X × X . But this can be easily shown, in fact for every n ∈ N it holds Z d (cid:0) ( x, y ) , ( x , y ) (cid:1) d π n ( x, y ) = Z (cid:2) d ( x, x ) + d ( y, y ) (cid:3) d π n ( x, y )= Z d ( x, x ) d µ ( x ) + Z d ( y, y ) d ν ( y )= Z d (cid:0) ( x, y ) , ( x , y ) (cid:1) dΠ( µ, ν )( x, y ) .
12n the other hand, taking π ∈ OptPlans d ( µ, ν ), condition (4) allows to use the dominatedconvergence theorem and deducelim sup n →∞ ( W d n ) ( µ, ν ) ≤ lim sup n →∞ Z d n d π = Z d d π = W ( µ, ν ) . Moreover for every compact set K ⊂ X × X there exists a continuous function φ K : X × X → [0 , φ K = 0 outside a compact set K ′ and f K ≡ K . Then φ K d n → φ K d uniformly,therefore lim inf n →∞ ( W d n ) ( µ, ν ) ≥ lim inf n →∞ Z φ K d n d π n = Z φ K d d π ≥ Z K d d π. Since K is arbitrary it is possible to conclude thatlim inf n →∞ ( W d n ) ( µ, ν ) ≥ Z d d π = W ( µ, ν ) , and consequently that lim n →∞ ( W d n ) ( µ, ν ) = W ( µ, ν ). Having that W ( π n , Π( µ, ν )) → n →∞ ( W d n ) ( µ, ν ) = W ( µ, ν ), the proof of Proposition 2.6 can be repeated step by stepand gives the thesis. (cid:3) Remark . This section has shown how the existence of a consistent geodesic flow and a con-sistent plan selection associated to it, can help in proving the very strict CD condition. However,I have not stated any results (except for Proposition 2.5) that would guarantee the existenceof these two objects in a metric measure space. To this aim, it would be very interesting toinvestigate under which assumptions on a given consistent geodesic flow G (or on the metricmeasure space), there exists a consistent plan selection associated to G . In the next section I willshow how a (double) minimization procedure allows to identify a consistent plan selection in aparticular metric measure space. It is possible that these arguments can also apply to a moregeneral context. R N The aim of this section is to prove the very strict CD (0 , ∞ ) condition for R N equipped with acrystalline norm and with the Lebesgue measure, using the theory developed in the last sectionand in particular Proposition 2.8. Let me point out that the Optimal Transport problem in theseparticular metric spaces has been already studied by Ambrosio Kirchheim and Pratelli in [6].They were able to solve the L -Monge problem using a secondary variational minimization inorder to suitably decompose the space in transport rays. Despite the problem I want to face andthe way I will do it are different from the theory developed in [6], I will in turn use a secondaryvariational problem to select a suitable transport plan connecting two given measures, obtaining,as a byproduct, the existence of optimal transport map between them.Before going on, I fix the notation I will use in this section. Given a finite set of vectors˜ V ⊂ R N such that span( ˜ V ) = R N , introduce the associate crystalline norm, which is defined asfollows k x k := max v ∈ ˜ V |h x, v i| and the corresponding distance d ( x, y ) := k x − y k = max v ∈ ˜ V |h x − y, v i| . k x k := max v ∈V h x, v i , d ( x, y ) := k x − y k = max v ∈V h x − y, v i , where V denotes the set ˜ V ∪ ( − ˜ V ).As the reader can easily guess, in this framework the choice of a consistent geodesic flow isnot really problematic, in fact it is sufficient to consider the Euclidean one, that is G : R N × R N → C ([0 , , R N )( x, y ) ( t (1 − t ) x + ty ) . The rest of the chapter will be then dedicated to the choice of a suitable plan selection, associ-ated to G , satisfying the requirements of Proposition 2.8. It will be identified via a secondaryvariational minimization. This type of procedure turns out to be useful in many situation (seefor example Chapter 2 and 3 in [14]) and in this specific case is inspired by the work of Rajala[12]. Let me now go into the details. Given two measures µ, ν ∈ P ( R N ), consider the usualKantorovich problem with cost c ( x, y ) = d ( x, y ), that ismin π ∈ Γ( µ,ν ) Z R N × R N d ( x, y ) d π ( x, y ) , calling Π ( µ, ν ) the set of its minimizers. Consequently consider the secondary variational problemmin π ∈ Π ( µ,ν ) Z R N × R N d eu ( x, y ) d π ( x, y ) , (5)where I denote by d eu the Euclidean distance, and denote by Π ( µ, ν ) ⊆ Π ( µ, ν ) the set ofminimizers, which can be easily seen to be not empty. In Theorem 3.2 I will show that, if µ isabsolutely continuous, Π ( µ, ν ) consists of a single element, but, in order to do this I have topreliminarily exploit the cyclical monotonicity properties of the plans in Π ( µ, ν ). Proposition 3.1.
Every π ∈ Π ( µ, ν ) is concentrated in a set Γ , such that for every ( x, y ) , ( x ′ , y ′ ) ∈ Γ it holds that d ( x, y ) + d ( x ′ , y ′ ) ≤ d ( x, y ′ ) + d ( x ′ , y ) , (6) moreover, if d ( x, y ) + d ( x ′ , y ′ ) = d ( x, y ′ ) + d ( x ′ , y ) , then d eu ( x, y ) + d eu ( x ′ , y ′ ) ≤ d eu ( x, y ′ ) + d eu ( x ′ , y ) . (7) Proof.
Fix π ∈ Π ( µ, ν ) and notice that, since in particular π ∈ Π ( µ, ν ), Proposition 1.6 yieldsthat π is concentrated in a set Γ satisfying (6). Furthermore, according to Proposition 1.7 andRemark 1.8, fix an upper semicontinuous Kantorovich potential φ for the cost c ( x, y ) = d ( x, y ),such that also φ c is upper semicontinuous. In particular for every η ∈ Π ( µ, ν ), it holds φ ( x ) + φ c ( y ) = c ( x, y ) = d ( x, y ) , for η -almost every ( x, y ) ∈ R N × R N . As a consequence, notice that being a minimizer of the secondary variational problem (5) isequivalent to realize the minimum ofmin η ∈ Π( µ,ν ) Z R N × R N ˜ c ( x, y ) d η ( x, y ) , where the cost ˜ c is defined as˜ c ( x, y ) = ( d eu ( x, y ) if φ ( x ) + φ c ( y ) = d ( x, y )+ ∞ otherwise . φ and φ c are upper semicontinuous, the cost ˜ c is lower semicontinuous.Thus Proposition 1.6 ensures that π is concentrated in a set Γ which is ˜ c -cyclically monotone.Moreover, up to modify Γ in a π -null set, it is possible to assume that for every ( x, y ) ∈ Γ φ ( x ) + φ c ( y ) = c ( x, y ) = d ( x, y ) . Now take ( x, y ) , ( x ′ , y ′ ) ∈ Γ with d ( x, y ) + d ( x ′ , y ′ ) = d ( x, y ′ ) + d ( x ′ , y ) and deduce that φ ( x ) + φ c ( y ) + φ ( x ′ ) + φ c ( y ′ ) = d ( x, y ) + d ( x ′ , y ′ ) = d ( x, y ′ ) + d ( x ′ , y ) . On the other hand φ ( x ) + φ c ( y ′ ) ≤ d ( x, y ′ ) and φ ( x ′ ) + φ c ( y ) ≤ d ( x ′ , y ), therefore I obtain φ ( x ) + φ c ( y ′ ) = d ( x, y ′ ) and φ ( x ′ ) + φ c ( y ) = d ( x ′ , y ) . Finally the ˜ c -cyclical monotonicity allows to conclude that d eu ( x, y ) + d eu ( x ′ , y ′ ) = ˜ c ( x, y ) + ˜ c ( x ′ , y ′ ) ≤ ˜ c ( x, y ′ ) + ˜ c ( x ′ , y ) = d eu ( x, y ′ ) + d eu ( x ′ , y ) , which is exactly (7). Summing up, it is easy to check that the set Γ = Γ ∩ Γ satisfies therequirements of Proposition 3.1. (cid:3) I can now go into the proof of one of the main results of this work.
Theorem 3.2.
Given two measures µ, ν ∈ P ( R N ) with µ absolutely continuous with respectto L n , there exists a unique π ∈ Π ( µ, ν ) and it is induced by a map. Proof.
Reasoning as in Remark 1.2, it is sufficient to prove that every plan in Π ( µ, ν ) is inducedby a map. So take π ∈ Π ( µ, ν ), applying Proposition 3.1 it is possible to find a full π -measure setΓ, satisfying the monotonicity requirements (6) and (7). Assume by contradiction that π is notinduced by a map, calling ( π x ) x ∈ R N ⊂ P ( R N ) the disintegration with respect to the projectionmap p , then π x is not a delta measure for a µ -positive set. Moreover, given a non-empty set V ⊆ V , define the sets˜ A z,V := (cid:8) x ∈ R N : d ( z, x ) = h z − x, v i for every v ∈ V (cid:9) ,A z,V := (cid:8) x ∈ ˜ A z,V : d ( z, x ) > h z − x, v i for every v ∈ V \ V (cid:9) ,A εz,V := (cid:8) x ∈ ˜ A z,V : d ( z, x ) > h z − x, v i + ε for every v ∈ V \ V (cid:9) . Notice that, for every fixed z ∈ R N , the sets A z,V constitute a partition of R N as V ⊆ V varies.Consequently, I divide the proof in three steps, whose combination will allow me to conclude bycontradiction. Step 1: Given two nonempty sets V , V ⊆ V such that v = v for every v ∈ V and v ∈ V (that is V ∩ V = ∅ ), the set E := (cid:8) z ∈ R N : π z ( A z,V ) > and π z ( A z,V ) > (cid:9) has zero µ -measure. First of all, notice that if E is non-empty, then for every fixed z ∈ R N there exist x ∈ A z,V and y ∈ A z,V such that d ( z, x ) = d ( z, y ) = 1 and in particular h x, v i = 1 > h y, v i for every v ∈ V and h y, v i = 1 > h x, v i for every v ∈ V . v = x − y , it holds that h ¯ v, v i > v ∈ V , h ¯ v, v i < v ∈ V . (8)Now, assume by contradiction that E has positive µ -measure, in particular it is non-emptyand there exists ¯ v satisfying (8). Moreover, notice that, since Γ is π -measurable and has fullmeasure, then Γ z := { ( z, y ) ∈ Γ : y ∈ R N } is π z -measurable with π z (Γ z ) = 1 for µ -almost every z ∈ R N . In particular for ε > E ε := (cid:8) z ∈ R N : π z ( A εz,V ∩ Γ z ) > π z ( A εz,V ∩ Γ z ) > (cid:9) has positive µ -measure, and thus it also has positive L N -measure. Take a Lebesgue densitypoint ¯ z of E ε , then in a neighborhood of ¯ z there exist z such that z, z + ǫ ¯ v ∈ E ε for a suitable0 < ǫ < ε k ¯ v k . Now, there exist x ∈ A εz,V and y ∈ A εz + ǫ ¯ v,V such that ( z, x ) , ( z + ǫ ¯ v, y ) ∈ Γ. Noticethat for every v ∈ V , it holds h x − ( z + ǫ ¯ v ) , v i = h x − z, v i − ǫ h ¯ v, v i < h x − z, v i = d ( z, x ) , (9)while for every w ∈ V \ V it is possible to perform the following estimate: h x − ( z + ǫ ¯ v ) , w i = h x − z, w i − ǫ h ¯ v, w i < d ( x, z ) − ε + ǫ k ¯ v k < d ( z, x ) . (10)The combination of (9) and (10) yields d ( x, z + ǫ ¯ v ) < d ( z, x ) . (11)Similarly, it holds h y − z, v i = h y − ( z + ǫ ¯ v ) , v i + ǫ h ¯ v, v i < h y − ( z + ǫ ¯ v ) , v i = d ( z + ǫ ¯ v, y ) , for every v ∈ V , and h y − z, w i = h y − ( z + ǫ ¯ v ) , w i + ǫ h ¯ v, w i < d ( z + ǫ ¯ v, y ) − ε + ǫ k ¯ v k < d ( z + ǫ ¯ v, y ) , for every w ∈ V \ V , which together show that d ( z, y ) < d ( z + ǫ ¯ v, y ) . (12)Now, the inequalities (11) and (12) allow to infer that d ( z, x ) + d ( z + ǫ ¯ v, y ) > d ( z, y ) + d ( z + ǫ ¯ v, x ) , contradicting the condition (6) of Proposition 3.1. Step 2: Given two nonempty sets V , V ⊆ V such that V ∩ V = ∅ and V = V , the set E := (cid:8) z ∈ R N : π z ( A z,V ) > and π z ( A z,V ) > (cid:9) has zero µ -measure. Call V = V ∩ V , W = V \ V and W = V \ V . Assume by contradiction that E has positive µ -measure, then for ε > E ε := (cid:8) z ∈ R N : π z ( A εz,V ) > π z ( A εz,V ) > (cid:9) has positive µ -measure too. As a consequence γ := Z E ε π z | A εz,V × π z | A εz,V d µ ( z )16s a strictly positive measure on R N × R N with γ (cid:0)(cid:8) ( x, x ) : x ∈ R N (cid:9)(cid:1) = 0. Thus there exists(¯ x, ¯ y ) ∈ supp( γ ) with ¯ x = ¯ y and then γ (cid:0) B δ (¯ x ) × B δ (¯ y ) (cid:1) > , for every δ >
0. In particular, proceeding as in the first step, it is possible to conclude that forevery δ > E δε := (cid:8) z ∈ R N : π z (cid:0) A εz,V ∩ Γ z ∩ B δ (¯ x ) (cid:1) > π z (cid:0) A εz,V ∩ Γ z ∩ B δ (¯ y ) (cid:1) > (cid:9) has positive µ -measure, and thus it also has positive L N -measure. Now, I divide the proof intwo cases, depending on the vector ¯ v = ¯ x − ¯ y : • Case 1: h ¯ v, v i = 0 for every v ∈ V .Since (¯ x, ¯ y ) ∈ supp( γ ), for every η > x η , y η , z η such that k ¯ x − x η k , k ¯ y − y η k < η and x η ∈ A εz η ,V , y η ∈ A εz η ,V . Then, given v ∈ V , for every v ∈ W it holds that h x η − z η , v i = h x η − z η , v i = h x η − ¯ x, v i + h ¯ v, v i + h ¯ y − y η , v i + h y η − z η , v i > h y η − z η , v i + ε , for η small enough. Thus, if η is sufficiently small, follows that h x η , v i > h y η , v i + ε v ∈ W , and similarly h x η , v i < h y η , v i − ε v ∈ W . Taking the limit as η →
0, clearly x η → ¯ x and y η → ¯ y , therefore I conclude that h ¯ x, v i > h ¯ y, v i and thus h ¯ v, v i > , for every v ∈ W , (13)and h ¯ x, v i < h ¯ y, v i and thus h ¯ v, v i < , for every v ∈ W . (14)Now, fix δ > h ¯ v, x i > h ¯ v, y i , for every x ∈ B δ (¯ x ) and y ∈ B δ (¯ y ) . (15)As already emphasized, the set E δε has positive Lebesgue measure, then take one of itsdensity points ¯ z . In a neighborhood of ¯ z there exists z , such that z, z + ǫ ¯ v ∈ E δε for a suitable0 < ǫ < ε k ¯ v k , subsequently take x ∈ A εz,V ∩ B δ (¯ x ) with ( z, x ) ∈ Γ, and y ∈ A εz + ǫ ¯ v,V ∩ B δ (¯ y )with ( z + ǫ ¯ v, y ) ∈ Γ. Notice that for every v ∈ V it holds h x − ( z + ǫ ¯ v ) , v i = h x − z, v i = d ( z, x ) , moreover (13) ensures that for every v ∈ W h x − ( z + ǫ ¯ v ) , v i < h x − z, v i = d ( z, x ) , while for every w ∈ V \ V the following estimate can be performed h x − ( z + ǫ ¯ v ) , w i = h x − z, w i − ǫ h ¯ v, w i < d ( x, z ) − ε + ǫ k ¯ v k < d ( z, x ) . d ( z + ǫ ¯ v, x ) = d ( z, x ) , (16)and analogously using (14) it can be proven that d ( z, y ) = d ( z + ǫ ¯ v, y ) . (17)On the other hand, the choice of δ I made (see (15)) guarantees that d eu ( z + ǫ ¯ v, x ) + d eu ( z, y ) = h z + ǫ ¯ v − x, z + ǫ ¯ v − x i + h z − y, z − y i = h z − x, z − x i + 2 h z − x, ǫ ¯ v i + h ǫ ¯ v, ǫ ¯ v i + h z − y, z − y i < h z − x, z − x i + h ǫ ¯ v, ǫ ¯ v i + 2 h z − y, ǫ ¯ v i + h z − y, z − y i = h z − x, z − x i + h z + ǫ ¯ v − y, z + ǫ ¯ v − y i = d eu ( z, x ) + d eu ( z + ǫ ¯ v, y ) , which, together with (16) and (17), contradicts the condition (7) of Proposition 3.1. • Case 2: there exists ¯ w ∈ V such that h ¯ v, ¯ w i 6 = 0.Without losing generality I can assume h ¯ v, ¯ w i >
0, then it is possible to fix a sufficientlysmall δ > η >
0, it holds h ¯ w, x i > h ¯ w, y i + η, for every x ∈ B δ (¯ x ) and y ∈ B δ (¯ y ) . Fix a vector ˜ v ∈ A z,V . Repeating the argument used in Case 1 it is possible to find apoint z ∈ R n , such that z, z + ǫ ˜ v ∈ E δε for a suitable 0 < ǫ < max (cid:8) ε k ˜ v k , η k ˜ v k (cid:9) . Then take x ∈ A εz,V ∩ B δ (¯ x ) and y ∈ A εz + ǫ ˜ v,V ∩ B δ (¯ y ) with ( z, x ) , ( z + ǫ ˜ v ) ∈ Γ, and notice that forevery v ∈ V it holds that h x − ( z + ǫ ˜ v ) , v i = h x − z, v i − ǫ h ˜ v, v i = d ( z, x ) − ǫ k ˜ v k while for every w ∈ V \ V I have h x − ( z + ǫ ˜ v ) , w i = h x − z, w i − ǫ h ˜ v, w i < d ( z, x ) − ε + ǫ k ˜ v k < d ( z, x ) − ǫ k ˜ v k , therefore follows that d ( z + ǫ ˜ v, x ) = d ( z, x ) − ǫ k ˜ v k . (18)On the other hand, observe that d ( z + ǫ ˜ v, y ) = h y − ( z + ǫ ˜ v ) , ¯ w i = h y − z, ¯ w i − ǫ h ˜ v, ¯ w i < d ( z, x ) − η + ǫ k ˜ v k < d ( z, x ) . (19)It is then possible to conclude that d ( z + ǫ ˜ v, x ) + d ( z, y ) ≤ (cid:0) d ( z, x ) − ǫ k ˜ v k (cid:1) + (cid:0) d ( z + ǫ ˜ v, y ) + ǫ k ˜ v k (cid:1) = d ( z, x ) + d ( z + ǫ ˜ v, y ) − ǫ k ˜ v k (cid:0) d ( z, x ) − d ( z + ǫ ˜ v, y ) (cid:1) < d ( z, x ) + d ( z + ǫ ˜ v, y ) , where I used both (18) and (19). This last inequality contradicts condition (6) of Proposition3.1. 18 tep 3: Given a nonempty set V ⊆ V , the set E := (cid:8) z ∈ R n : π z | A z,V is not a delta measure (cid:9) has zero µ -measure. The proof of this step is very similar to the one of Step 2, nevertheless I decided to presentit anyway, but avoiding all the details which can be easily fixed following the proof of Step 2.Assume by contradiction that E has positive µ -measure, then for ε > E ε := (cid:8) z ∈ R n : π z | A εz,V is not a delta measure (cid:9) has positive µ -measure too. As a consequence γ := Z E ε π z | A εz,V × π z | A εz,V d µ ( z )is a strictly positive measure on R N × R N that is not concentrated on (cid:8) ( x, x ) : x ∈ R N (cid:9) . Thusthere exists (¯ x, ¯ y ) ∈ supp( γ ) with ¯ x = ¯ y and then γ (cid:0) B δ (¯ x ) × B δ (¯ y ) (cid:1) > , for every δ >
0. In particular, proceeding as in the first step, it is possible to conclude that forevery δ > E δε := (cid:8) z ∈ R N : π z (cid:0) A εz,V ∩ Γ z ∩ B δ (¯ x ) (cid:1) > π z (cid:0) A εz,V ∩ Γ z ∩ B δ (¯ y ) (cid:1) > (cid:9) has positive µ -measure, and thus it also has positive L N -measure. Now, as I did in Step 2, Idivide the proof in two cases: • Case 1: h ¯ v, v i = 0 for every v ∈ V .First of all, fix δ > h ¯ v, x i > h ¯ v, y i , for every x ∈ B δ (¯ x ) and y ∈ B δ (¯ y ) . Proceeding as in Step 2, I can find z ∈ R n , such that z, z + ǫ ¯ v ∈ E δε for a positive, suitablysmall ǫ . Subsequently take x ∈ A εz,V ∩ B δ (¯ x ) with ( z, x ) ∈ Γ, and y ∈ A εz + ǫ ¯ v,V ∩ B δ (¯ y ) with( z + ǫ ¯ v, y ) ∈ Γ. Following the proof of Step 2, it is easy to realize that d ( z + ǫ ¯ v, x ) = d ( z, x ) , (20)and d ( z, y ) = d ( z + ǫ ¯ v, y ) . (21)On the other hand, the choice of δ I made guarantees that d eu ( z + ǫ ¯ v, x ) + d eu ( z, y ) < d eu ( z, x ) + d eu ( z + ǫ ¯ v, y ) , which, together with (20) and (21), contradicts the condition (7) of Proposition 3.1. • Case 2: there exists ¯ w ∈ V such that h ¯ v, ¯ w i 6 = 0.Without losing generality I can assume h ¯ v, ¯ w i >
0, then it is possible to fix a sufficientlysmall δ > η >
0, it holds that h ¯ w, x i > h ¯ w, y i + η, for every x ∈ B δ (¯ x ) and y ∈ B δ (¯ y ) . v ∈ A z,V , it is possible to find a point z ∈ R n , such that z, z + ǫ ˜ v ∈ E δε for a positive, suitably small ǫ . Then take x ∈ A εz,V ∩ B δ (¯ x ) and y ∈ A εz + ǫ ˜ v,V ∩ B δ (¯ y ) with( z, x ) , ( z + ǫ ˜ v ) ∈ Γ. Proceeding as I did in Step 2, it is easy to notice that d ( z + ǫ ˜ v, x ) = d ( z, x ) − ǫ k ˜ v k . (22)and d ( z + ǫ ˜ v, y ) < d ( z, x ) . (23)Then, combining (22) and (23), I can conclude that d ( z + ǫ ˜ v, x ) + d ( z, y ) < d ( z, x ) + d ( z + ǫ ˜ v, y ) , contradicting condition (6) of Proposition 3.1.As anticipated before, it is easy to realize that the combination of the three steps allows toconclude the proof. (cid:3) At this point it is clear that Theorem 3.2 provides a plan selection on P ac ( R N ) × P ac ( R N ),simply imposing Π( µ, ν ) to be equal to the only optimal transport plan in Π ( µ, ν ). The followingproposition ensures that Π is a consistent plan selection. Proposition 3.3.
The map Π is a consistent plan selection, associated to G . Proof.
Considering how Π has been defined, in order to conclude the proof, is sufficient to proveconditions 2.1 and 2.2 of Lemma 2.3. It is easy to realize that condition 2.1 is satisfied since f · Π( µ, ν ) ≪ Π( µ, ν ) with bounded density, for every suitable f . Condition 2.2 is a little bittrickier and I am going to prove it with full details.Assume by contradiction that, for some µ, ν ∈ P ac ( R N ), π := ( G s , G t ) Π( µ, ν ) is nota minimizer for the secondary variational problem (5), with absolutely continuous marginals µ s := ( G s ) Π( µ, ν ) and µ t := ( G t ) Π( µ, ν ). Since π is clearly an optimal transport plan, thismeans that there exists π ∈ OptPlans( µ s , µ t ) such that Z d eu ( x, y ) d π < Z d eu ( x, y ) d π . Then Dudley’s gluing lemma ensures the existence of a probability measure ˜ π ∈ P (( R N ) ) suchthat ( p , p ) ˜ π = π , ( p , p ) ˜ π = π and ( p , p ) ˜ π = π , where π := ( G , G s ) Π( µ, ν ) and π := ( G t , G ) Π( µ, ν ). Defining ¯ π := ( p , p ) ˜ π it is possibleto perform the following estimate Z d ( x, y ) d¯ π ( x, y ) = Z d ( x, y ) d˜ π ( x, z, w, y ) ≤ Z (cid:0) d ( x, z ) + d ( z, w ) + d ( w, y ) (cid:1) d˜ π ( x, z, w, y )= Z d ( x, z ) d π + Z d ( z, w ) d π + Z d ( w, y ) d π + 2 Z d ( x, z ) d ( z, w ) d˜ π ( x, z, w, y ) + 2 Z d ( x, z ) d ( w, y ) d˜ π ( x, z, w, y )+ 2 Z d ( z, w ) d ( w, y ) d˜ π ( x, z, w, y ) . Z d ( x, z ) d ( z, w ) d˜ π ( x, z, w, y ) = s ( t − s ) Z (cid:18) s d ( x, z ) (cid:19)(cid:18) t − s d ( z, w ) (cid:19) d˜ π ( x, z, w, y ) ≤ t − ss Z d ( x, z ) d π + st − s Z d ( z, w ) d π and similarly2 Z d ( x, z ) d ( w, y ) d˜ π ( x, z, w, y ) ≤ − ts Z d ( x, z ) d π + s − t Z d ( w, y ) d π, Z d ( z, w ) d ( w, y ) d˜ π ( x, z, w, y ) ≤ − tt − s Z d ( z, w ) d π + t − s − t Z d ( w, y ) d π. Putting together this last three inequalities, it is possible to deduce that Z d ( x, y ) d¯ π ( x, y ) ≤ s Z d ( x, z ) d π + 1 t − s Z d ( z, w ) d π + 11 − t Z d ( w, y ) d π = 1 s W ( µ, µ s ) + 1 t − s W ( µ s , µ t ) + 11 − t W ( µ t , ν ) = W ( µ, ν ) , where I used the fact that G Π( µ, ν ) is an optimal geodesic plan. In particular this shows that¯ π ∈ OptPlans( µ, ν ). Furthermore, performing the same computation as before, one can infer that Z d eu ( x, y ) d¯ π ( x, y ) ≤ s Z d eu ( x, z ) d π + 1 t − s Z d eu ( z, w ) d π + 11 − t Z d eu ( w, y ) d π < s Z d eu ( x, z ) d π + 1 t − s Z d eu ( z, w ) d π + 11 − t Z d eu ( w, y ) d π = Z d eu ( x, y ) dΠ( µ, ν ) , where this last equality holds because G Π( µ, ν ) is concentrated in Euclidean geodesic. Noticethat I have found ¯ π ∈ OptPlans( µ, ν ) such that Z d eu ( x, y ) d¯ π ( x, y ) < Z d eu ( x, y ) dΠ( µ, ν ) , this contradicts the definition of Π. (cid:3) In order to deduce the main result of this section I only have to prove the approximationproperty stated in Proposition 2.8, and to this aim I need to preliminary state and prove thefollowing proposition. Let me also point out that this result can be proven using general theorems(see for example Theorem 10.27 in [21] or Theorem 1.3.1 in [7]), anyway I prefer to present aproof that uses only cyclical monotonicity arguments, similar to the ones explained previously.
Proposition 3.4.
Let N : R N → R + be a smooth norm, such that N : R N → R + is k -convex forsome k > . Calling d : R N × R N → R + the associated distance and given µ, ν ∈ P ( R N ) with µ ≪ L N , there exists a unique π ∈ OptPlans( µ, ν ) and it is induced by a map. Proof.
According to Remark 1.2, it is sufficient to prove that every π ∈ OptPlans( µ, ν ) it isinduced by a map. To this aim, fix π ∈ OptPlans( µ, ν ) and call Γ the π -full measure, d -cyclicallymonotone set, provided by Proposition 1.6. Assume by contradiction that π is not induced by amap, denote by { π x } x ∈ X the disintegration kernel with respect to the projection map p , then21 x is not a delta measure for a µ -positive set of x . Therefore there exists a compact set A ⊂ R N with µ ( A ) >
0, such that π x is not a delta measure for every x ∈ A . Consequently consider η := Z A π x × π x d µ, which is a positive measure on R N × R N . Moreover η is not concentrated on { ( x, x ) : x ∈ R N } ,thus there exists (¯ x, ¯ y ) ∈ supp( η ) with ¯ x = ¯ y and in particular η ( B δ (¯ x ) × B δ (¯ y )) > δ >
0. Now call v = ¯ y − ¯ x and notice that, since N is smooth there exists ¯ δ > z ∈ A it holds that (cid:12)(cid:12)(cid:12)(cid:12) ∂∂v N ( x − z ) − ∂∂v N (¯ x − z ) (cid:12)(cid:12)(cid:12)(cid:12) < k d (¯ y, ¯ x )for every x ∈ B δ (¯ x ), and (cid:12)(cid:12)(cid:12)(cid:12) ∂∂v N ( y − z ) − ∂∂v N (¯ y − z ) (cid:12)(cid:12)(cid:12)(cid:12) < k d (¯ y, ¯ x )for every y ∈ B δ (¯ y ). Moreover, since N is k -convex, for every z ∈ A it holds that ∂∂v N (¯ y − z ) ≥ ∂∂v N (¯ x − z ) + k d (¯ y, ¯ x ) , and consequently ∂∂v N ( y − z ) > ∂∂v N ( x − z ) (24)for every x ∈ B δ (¯ x ) and every y ∈ B δ (¯ y ). On the other hand, since η ( B ¯ δ (¯ x ) × B ¯ δ (¯ y )) >
0, theset A ¯ δ = { z ∈ R N : π z ( B ¯ δ (¯ x )) > π z ( B ¯ δ (¯ y )) > } . has positive µ -measure and thus it has positive L N -measure. Let ¯ z be the density point of A ¯ δ , then in a neighborhood ¯ z there exists z such that z, z + ǫv ∈ A ¯ δ for some 0 < ǫ < ¯ δ k v k .Consequently, it is possible to find x ∈ B ¯ δ (¯ x ) and y ∈ B ¯ δ (¯ y ), such that( z + ǫv, x ) , ( z, y ) ∈ Γ . Then it holds that d ( z, x ) + d ( z + ǫv, y ) = N ( x − z ) + N ( y − ( z + ǫv ))= N ( x − ( z + ǫv )) + Z ǫ ∂∂v N ( x − sv − z )) d s + N ( y − z ) − Z ǫ ∂∂v N ( y − sv − z ) d s< d ( z + ǫv, x ) + d ( z, y ) , where the last passage follows from (24). This last inequality contradicts the d -cyclical mono-tonicity of Γ, concluding the proof. (cid:3) Having a consistent geodesic flow and an associated plan selection, it only remains to applyProposition 2.8 and deduce the main result. In order to do so, I introduce a sequence ( d n ) n ∈ N ofdistances on R N by requiring the following three properties: • for every n, d n is induced by a smooth norm N , such that N is k -convex for some k > d n converges to d uniformly on compact sets, • n ( d n − d ) converges to d eu uniformly on compact sets, and n ( d n − d ) ≤ d eu for every n .It is easy to see that such a sequence exists. Now, fixed a pair of absolutely continuous measures µ, ν ∈ P ac ( R N ), Proposition 3.4 ensures that for every n there exists a unique transport plan π n between µ and ν , with respect to the cost c ( x, y ) = d n ( x, y ). Let me now prove that it is possibleto apply Proposition 2.8. Proposition 3.5.
The maps G and Π and the sequences ( d n ) and ( π n ) I introduced satisfy theassumptions of Proposition 2.8 with K = 0 . Proof.
Condition 2 is easily satisfied, in fact since d n is induced by a strictly convex norm the onlygeodesics in ( R N , π n ) are the Euclidean ones. Then, because π n is unique and Proposition 1.22holds, it is clear that the entropy functional is convex along G π n , with respect to the distance d n . Let me now prove condition 1. Notice that π n ∈ Γ( µ, ν ) for every n , therefore the sequence( π n ) is tight and Prokhorov theorem ensures the existence of π ∈ Γ( µ, ν ) such that, up to theextraction of a subsequence, π n ⇀ π . I am now going to prove that π ∈ Π ( µ, ν ). Observe that π n is an optimal transport plan for the distance d n and thus Z d n d π n ≤ Z d n d˜ π ∀ ˜ π ∈ Γ( µ, ν ) , therefore for every compact set C ⊂ R N it holds Z C d n d π n ≤ Z d n d˜ π ∀ ˜ π ∈ Γ( µ, ν ) . It is then possible to pass to the limit as n → ∞ , using the uniform convergence for the left handside and the dominated convergence (ensured by (4)) at the right hand side, obtaining Z C d d π ≤ Z d d˜ π ∀ ˜ π ∈ Γ( µ, ν ) . Since this last equation holds for every compact set C ⊂ R N , it is possible to conclude that Z d d π ≤ Z d d˜ π ∀ ˜ π ∈ Γ( µ, ν ) , in particular π ∈ Π ( µ, ν ). Using once more the minimizing property of π n , follows that Z d d˜ π + Z ( d n − d ) d π n ≤ Z d n d π n ≤ Z d n d˜ π = Z d d˜ π + Z ( d n − d ) d˜ π ∀ ˜ π ∈ Π ( µ, ν ) , consequently it holds that Z n ( d n − d ) d π n ≤ Z n ( d n − d ) d˜ π ∀ ˜ π ∈ Π ( µ, ν ) , and proceeding as before I can infer that Z d eu d π ≤ Z d eu d˜ π ∀ ˜ π ∈ Π ( µ, ν ) . In particular π ∈ Π ( µ, ν ) and this concludes the proof, considering the definition of the mapΠ. (cid:3) Corollary 3.6.
The metric measure space ( R N , d , L N ) is a very strict CD (0 , ∞ ) space andconsequently it is weakly essentially non-branching. Aknowlegments : This article contains part of the work I did for my master thesis, that wassupervised by Luigi Ambrosio and Karl-Theodor Sturm.
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