aa r X i v : . [ m a t h . M G ] J a n Example of an Highly Branching CD Space
Mattia Magnabosco
Abstract
In [3] Ketterer and Rajala showed an example of metric measure space, satisfying themeasure contraction property
MCP (0 , CD (0 , ∞ ) condition, proving the non-constancy of topological dimension for CD spaces. Thisexample also shows that the weak curvature dimension bound, in the sense of Lott-Sturm-Villani, is not sufficient to deduce any reasonable non-branching condition. Moreover, it allowsto answer to some open question proposed by Schultz in [7], about strict curvature dimensionbounds and their stability with respect to the measured Gromov Hausdorff convergence. In their remarkable works Lott, Villani [4] and Sturm [10, 11] introduced a weak notion ofcurvature dimension bounds, which strongly relies on the theory of Optimal Transport. Inspiredby some results that hold in the Riemannian case, they defined a consistent notion of curvaturedimension bound for metric measure spaces, that is known as CD condition. The metric measurespaces satisfying the CD condition are called CD spaces and enjoy some remarkable analytic andgeometric properties.In this work I present an example of an highly branching CD space, that shows how the weakcurvature dimension bound is not sufficient to deduce any type of non-branching condition. Thisexample is a refinement of the one by Ketterer and Rajala [3], in particular, the topological andmetric structure of the space will be essentially the same, while the reference measure will bemore complicated. The metric measure space considered by Ketterer and Rajala in [3] satisfiesthe so called measure contraction property
MCP (0 , • any reasonable non-branching condition does not hold in general for (weak) CD spaces, • the existence of an optimal transport map between two absolutely continuous marginals isnot granted in (weak) CD spaces, without assuming a non-branching condition, • the very strict CD condition studied by Schultz ([7],[8],[9]) is strictly stronger than the weakone, 1 the constancy of topological dimension does not hold in general for (weak) CD spaces, • the strict CD condition I will define in the last section is not stable with respect of themeasured Gromov Hausdorff convergence, • the strict CD condition is strictly stronger than the weak one.Let me now briefly explain the structure of this work. In the first section I recall somepreliminary results regarding both the basis of the Optimal Transport theory and CD spaces. Inthe second section I am going to introduce the metric measure spaces which will be the subjectof the rest of the article. In section 3 I simply state and prove some algebraic lemma, whichwill help me doing the subsequent computations. In the fourth section I present some result byRajala [5], related to Jacobi equation and how it can be used to prove entropy convexity. Thistheory requires the existence of a suitable midpoint selection map, which will be introduced inSection 5, where I will also prove some of its properties. The sixth section contains the proof ofthe main theorem, that puts together all the results proven so far. The last section aims to drawall the conclusions listed before. This first section aims to collect all the preliminary results this work needs in order to be selfcontained. In particular I am going to introduce the Wasserstein space and the entropy functionalon it, being then able to define the notions of curvature dimension bound and CD space. MoreoverI am going to briefly discuss the relation between curvature dimension bound and non-branchingconditions, that is one of the main motivation for this work. Finally I will define the measuredGromov Hausdorff convergence of metric measure spaces, stating in the end the stability ofcurvature dimension bounds with respect to this convergence.
Denote by P ( X ) the set of Borel probability measures on a Polish metric space ( X, d ). Giventwo measures µ, ν ∈ P ( X ) and a Borel cost function c : X × X → [0 , ∞ ], the Optimal Transportproblem asks to find minima and minimizers of the quantitymin Z X × Y c ( x, y ) d π ( x, y ) , (1)where π varies among all probability measures in P ( X × X ) with first marginal equal to µ andsecond marginal equal to ν . If the cost function c is lower semicontinuous, the minimum in (1) isattained. The minimizers of this problem are called optimal transport plans and the set of all ofthem will be denoted by OptPlans( µ, ν ). An optimal transport plan π ∈ OptPlans( µ, ν ) is saidto be induced by a map if there exists a µ -measurable map T : X → X so that π = (id , T ) µ ,such a map T will be called optimal transport map.A fundamental approach in facing the Optimal Transport problem is the one of c -duality, whichallows to prove some very interesting and useful results. Below I report only the most basicstatement, which is the only result I will need in this work. Definition 1.1.
A set Γ ⊂ X × X 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 .2 roposition 1.2. Let X be a Polish space and c : X × X → [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. From now on I am going to consider the Optimal Transport problem in the special case inwhich the cost function is equal to the distance squared, that is c ( x, y ) = d ( x, y ). In this contextthe minimization problem induces the so called Wasserstein distance on the space P ( X ) ofprobabilities with finite second order moment, that is P ( X ) := (cid:26) µ ∈ P ( X ) : Z d ( x, x ) d µ ( x ) < ∞ for one (and thus all) x ∈ X (cid:27) . Definition 1.3 (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) . It is easy to realize that W is actually a distance on P ( X ), moreover ( P ( X ) , W ) is a Polishmetric space.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 ). First of all, notice that every measure π ∈ P ( C ([0 , , X )) induces a curve [0 , ∋ t → µ t = ( e t ) π ∈ P ( X ), therefore in the followingI will consider measures in P ( C ([0 , , X )) in order to consider curves in the Wasserstein space. Proposition 1.4. If ( X, d ) is a geodesic space than ( P ( X ) , W ) is geodesic as well. In partic-ular, given two measures µ, ν ∈ P ( X ) , the measure π ∈ P ( C ([0 , , X )) is a constant speedWassertein 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 ) . In this subsection I introduce the notions of curvature dimension bound and CD space, pioneeredby Lott and Villani [4] and Sturm [10, 11]. Their definition relies on the notion of entropyfunctional. As it will be soon clear, the most appropriate framework in which deal with theentropy functional, is the one of metric measure spaces.
Definition 1.5.
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.In this work I will only deal with compact metric measure spaces, and in particular m ( X ) < ∞ .Let me now properly define the entropy functional. Definition 1.6.
In a metric measure space ( X, d , m ), define the relative entropy functional withrespect to the reference measure m Ent : P ( X ) → R ∪ { + ∞} asEnt( µ ) := (R ρ log ρ d m if µ ≪ ν and µ = ρ m + ∞ otherwise . In this context of this work the entropy functional Ent is lower semicontinuous with respect to theWasserstein convergence, this is not always true in the non-compact case, when it might happenthat m ( X ) = + ∞ .I can now give the definitions of CD condition and CD space.3 efinition 1.7. 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 absolutely continuous measures µ , µ ∈ P ( X ) there exists a Wasserstein geodesic with constant speed ( µ t ) t ∈ [0 , ⊂ P ( X ) connectingthem, along which the relative entropy 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 , . (2)Moreover ( X, d , m ) is said to satisfy a strong CD ( K, ∞ ) condition and to be a strong CD ( K, ∞ )space if, for every absolutely continuous measures µ , µ ∈ P ( X ), the relative entropy functionalis K -convex along every Wasserstein geodesic with constant speed connecting them.Let me also state a very useful proposition, which provides a simple strategy to prove the (weak) CD ( K, ∞ ) condition. Its proof can be found in [10], anyway I present a brief sketch of it, in orderto be self contained. Proposition 1.8.
The metric measure space ( X, d , m ) is a CD ( K, ∞ ) space if for every pair ofabsolutely continuous measures µ , µ ∈ P ( X ) there exists a midpoint η ∈ P ( X ) of µ and µ , absolutely continuous with respect to m , satisfying Ent( η ) ≤
12 Ent( µ ) + 12 Ent( µ ) − K W ( µ , µ ) . (3) Proof.
Given two absolutely continuous measures µ , µ ∈ P ( X ), define µ / as a midpoint of µ and µ satisfying (3). Similarly define µ / as a midpoint of µ and µ / and µ / as a µ / and µ , both satisfying (3). Proceeding in this way, it is possible to define µ t for all dyadic times t ∈ (cid:8) k h : h ∈ N + , k = 1 , . . . , h − (cid:9) . An easy induction argument on h shows thatEnt( µ t ) ≤ (1 − t ) Ent( µ ) + t Ent( µ ) − t (1 − t ) K W ( µ , µ ) . for every dyadic time t ∈ { k h : h ∈ N + , k = 1 , . . . , h − } . Defining the geodesic ( µ t ) t ∈ [0 , as thecontinuous extension, the lower semicontinuity of the entropy ensures that, for every t ∈ [0 , µ t satisfies the equation (2). (cid:3) In the last part of this subsection I want to present the relation between curvature dimensionbounds and non-branching conditions. The most important result in this context was proven byRajala and Sturm in [6]:
Theorem 1.9.
Every strong CD ( K, ∞ ) metric measure space ( X, d , m ) is essentially non-branching,that is for every absolutely continuous measures µ , µ ∈ P ( X ) , every optimal geodesic planconnecting them is concentrated on a non-branching set of geodesics. The work of Rajala and Sturm was then generalized by Schultz [7] to the context of very strictCD spaces.
Definition 1.10.
A metric measure space ( X, d , m ) is called a very strict CD ( K, ∞ ) space iffor every absolutely continuous measures µ , µ ∈ P ( X ) there exists an optimal geodesic plan η ∈ OptGeo( µ , µ ), so that the entropy functional Ent satisfies the K-convexity inequality along(restr t t ) ( f η ) for every t < t ∈ [0 , f : Geo( X ) → R + with R f d η = 1.As the reader can easily notice, these spaces are not in general essentially non-branching, butthey satisfy a weaker condition that I will call weak essentially non-branching .4 efinition 1.11 (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 1.12 (Schultz [7]) . Every very strict CD ( K, ∞ ) space is weakly essentially non-branching. Notice that the very strict CD condition is intermediate between the weak and the strong one.It is easy to find examples of very strict CD spaces, which are not strong CD spaces, while it isnot obvious if the very strict condition is strictly stronger than the weak one. In this work I amgoing to present an example of an highly branching (weak) CD space which is not very strict CD.
In this subsection I introduce (following [12]) a notion of convergence for metric measure spaces,that is called measured Gromov Hausdorff convergence. Roughly speaking, it is the combinationof Hausdorff topology for the metric side, and weak topology for the measure side. In order toproperly define the measured Gromov Hausdorff convergence I have to preliminary introduce thenotion of ε -isometry. Definition 1.13.
A measurable map f : ( X, d , m ) → ( X ′ , d ′ , m ′ ) between two metric measurespaces is called an ε -isometry if1. it almost preserves the distances, that is: (cid:12)(cid:12) d (cid:0) f ( x ) , f (cid:0) x ′ (cid:1)(cid:1) − d (cid:0) x, x ′ (cid:1)(cid:12)(cid:12) ≤ ε for every x, x ′ ∈ X,
2. it is almost surjective, that is: ∀ y ∈ X ′ , there exists x ∈ X such that d ( f ( x ) , y ) ≤ ε. Definition 1.14.
Let ( X k , d k , m k ) k ∈ N and ( X, d , m ) be compact metric measure spaces. It issaid that the sequence ( X k , d k , m k ) k ∈ N converges to ( X, d , m ), in the measured Gromov Hausdorffsense, if for every k there exists a measurable ε k -isometry f k : X k → X , where ε k →
0, such that( f k ) m k ⇀ m as k → ∞ . (4)The measured Gromov Hausdorff convergence can be in some sense metrized by the D distance,introduced by Sturm in [11]. Moreover in [2] Gigli, Mondino and Savar´e showed that somedifferent notion of convergence for (pointed) metric measure spaces are equivalent to the (pointed)measured Gromov Hausdorff convergence.I end this subsection, stating the stability of the (weak) CD condition with respect to the measuredGromov Hausdorff convergence. Theorem 1.15.
Let ( X k , d k , m k ) k ∈ N be a sequence of compact metric measure spaces convergingin the measured Gromov Hausdorff sense to a compact metric measure space ( X, d , m ) . Given K ∈ R , if each ( X k , d k , m k ) k ∈ N satisfies the weak curvature dimension condition CD ( K, ∞ ) , thenalso ( X, d , m ) satisfies CD ( K, ∞ ) . In this section I am going to introduce the metric measure spaces that will be studied in the restof this work. The definitions that follow are actually involved and quite complicated, thus I invitethe reader to look at Figure 1 and Figure 2, in order to better understand.5 f k,ε ( x )Figure 1: The metric measure space ( X k,ε , d ∞ , m k,K,ε ) with ε > Definition 2.1.
Fix ε, k ∈ R such that 0 ≤ ε < k < and let ϕ : R → [0 ,
1] be a continuousfunction such that R ϕ = k , ϕ ≤ R and ϕ = 0 outside [ − k, k ]. Define the function f k,ε : [ − , → R + prescribing f ′′ k,ε ( x ) = ϕ ( x ) , f ′ k,ε ( −
1) = 0 , f k,ε ( −
1) = ε. Consequently define the set X k,ε = { ( x, y ) ∈ R : x ∈ [ − ,
1] and 0 ≤ y ≤ f k,ε ( x ) } . In the following I will use this notation: L = X k, ∩ (cid:0) { f k, = 0 } × R (cid:1) and C = X k, ∩ (cid:0) { f k, = 0 } × R (cid:1) , for sake of simplicity, I will not explicit the k dependence, because it will be clear from thecontext. Definition 2.2.
Given ε, k ∈ R with 0 < ε < k < and K ≥
1, define the measure m k,K,ε on X k,ε as m k,K,ε = m k,K,ε ( x, y ) · L | X k,ε := 1 f k,ε ( x ) exp − K (cid:18) yf k,ε ( x ) (cid:19) ! L | X k,ε . While for every 0 < k < and K ≥
1, define the measure m k,K, on X k, as m k,K, := C K · χ { f k, ( x )=0 } · H (cid:12)(cid:12)(cid:12) { y =0 } + χ { f k, ( x ) > } f k, ( x ) exp − K (cid:18) yf k, ( x ) (cid:19) ! · L | X k, , where C K = Z e − Ky d y. f k, ( x )Figure 2: The metric measure space ( X k, , d ∞ , m k,K, ).6otice that a simple change of variable shows that( p x ) m k,K, = ( p x ) m k,K,ε = C K · χ {− ≤ x ≤ } · H , for every suitable k , K and ε . Moreover, since I have imposed ε < k , it is easy to realize that f k,ε ( x ) < k for every x ∈ [ − ,
1] (see Figure 1).In the following I am going to prove that for suitable k and K the metric measure space( X k,ε , d ∞ , m k,K,ε ) is a CD (0 , ∞ ) space for every ε < k . In particular, in the next four sections I willconsider the metric measure space ( X k,ε , d ∞ , m k,K,ε ), but I will avoid to indicate the parameters k , K and ε at the subscript, in order to ease the notation. Later, in the last section, I am going toprove the measured Gromov Hausdorff convergence of the spaces ( X k,ε , d ∞ , m k,K,ε ) to the space( X k, , d ∞ , m k,K, ), as ε goes to 0. Combining this with the stability result (Theorem 1.15), willfollow that ( X k, , d ∞ , m k,K, ) is itself a CD (0 , ∞ ) space. In this section I simply state and prove four algebraic lemmas that will be fundamental in theproof of the main theorem. In particular they will only help in carrying on the computation, butthey do not hide any particular or sophisticated idea. For this reason I do not spend a lot ofwords on them and I immediately go through the proofs.
Lemma 3.1.
There exists a constant C such that for every A ∈ R + and every δ with | δ | < itholds
12 log( A ) ≤ log (cid:18) (cid:18)
12 + δ (cid:19) ( A − (cid:19) + Cδ (5) Proof.
Notice that for some A ∈ R + the inequality (5) holds also without the term + Cδ . Inparticular this is true if and only if √ A ≤ (cid:18)
12 + δ (cid:19) ( A − . (6)Through elementary computation it is easy to show that inequality (6) holds for every suitable δ if A / ∈ (cid:20) − | δ | (2 δ + 1) , | δ | (2 δ + 1) (cid:21) ⊂ [1 − | δ | , | δ | ] . Therefore, in order to conclude the proof, it is sufficient to prove inequality (5) for A ∈ [1 − | δ | , | δ | ]. In this caselog (cid:18) (cid:18)
12 + δ (cid:19) ( A − (cid:19) = log (cid:18) A −
1) + δ ( A − (cid:19) = log (cid:18) A − (cid:19) + Z δ ( A −
11 + ( A −
1) + t d t ≥ log (cid:18) A − (cid:19) − Z | δ ( A − | − | δ | − δ d t ≥ log (cid:18) A − (cid:19) − | δ ( A − |≥
12 log( A ) − δ , where the last inequality follows from the concavity of the logarithm. (cid:3) emma 3.2. Given ( x , y ) , ( x , y ) ∈ X , let γ : [0 , → X be the function: t (cid:18) (1 − t ) y f ( x ) + t y f ( x ) (cid:19) f ((1 − t ) x + tx ) then1. for every t ∈ [0 , it holds (cid:12)(cid:12)(cid:12)(cid:12) γ ′ ( t ) x − x − y − y x − x (cid:12)(cid:12)(cid:12)(cid:12) ≤ k,
2. if k < , for every t ∈ [0 , it holds (cid:12)(cid:12) γ ′′ ( t ) (cid:12)(cid:12) ≤ ( x − x ) kf ((1 − t ) x + tx ) (cid:18)(cid:12)(cid:12)(cid:12)(cid:12) y − y x − x (cid:12)(cid:12)(cid:12)(cid:12) + 2 (cid:19) . Proof.
The first derivative of γ is γ ′ ( t ) = (cid:18) y f ( x ) − y f ( x ) (cid:19) f ((1 − t ) x + tx ) + (cid:18) (1 − t ) y f ( x ) + t y f ( x ) (cid:19) f ′ ((1 − t ) x + tx )( x − x ) . Therefore, denoting x t = (1 − t ) x + tx , it holds γ ′ ( t ) x − x = y − y x − x + (cid:18) f ( x ) − f ( x t ) (cid:19) y x − x f ( x t ) + (cid:18) f ( x t ) − f ( x ) (cid:19) y x − x f ( x t )+ (cid:18) (1 − t ) y f ( x ) + t y f ( x ) (cid:19) f ′ ( x t )= y − y x − x + y f ( x ) f ( x t ) − f ( x ) x − x + y f ( x ) f ( x ) − f ( x t ) x − x + (cid:18) (1 − t ) y f ( x ) + t y f ( x ) (cid:19) f ′ ( x t ) . On the other hand, it is possible to perform the following estimate (cid:12)(cid:12)(cid:12)(cid:12) y f ( x ) f ( x t ) − f ( x ) x − x (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12) y f ( x ) (cid:12)(cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12) f ( x t ) − f ( x ) x − x (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12) f ( x t ) − f ( x ) x − x (cid:12)(cid:12)(cid:12)(cid:12) ≤ sup f ′ ≤ k, and the same calculation can be done for the symmetric term, thus (cid:12)(cid:12)(cid:12)(cid:12) y f ( x ) f ( x ) − f ( x t ) x − x (cid:12)(cid:12)(cid:12)(cid:12) ≤ k. Moreover, a similar procedure shows that (cid:12)(cid:12)(cid:12)(cid:12)(cid:18) (1 − t ) y f ( x ) + t y f ( x ) (cid:19) f ′ ( x t ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ k. Putting together all this estimates, it is possible to conclude that (cid:12)(cid:12)(cid:12)(cid:12) γ ′ ( t ) x − x − y − y x − x (cid:12)(cid:12)(cid:12)(cid:12) ≤ k. The second derivative of γ is γ ′′ ( t ) = 2 (cid:18) y f ( x ) − y f ( x ) (cid:19) f ′ ( x t )( x − x ) + (cid:18) (1 − t ) y f ( x ) + t y f ( x ) (cid:19) f ′′ ( x t )( x − x ) (cid:18) y f ( x ) − y f ( x ) (cid:19) f ′ ( x t )( x − x )= 2 f ′ ( x t )( x − x ) f ( x t ) (cid:20) y − y x − x + y f ( x ) f ( x t ) − f ( x ) x − x + y f ( x ) f ( x ) − f ( x t ) x − x (cid:21) . Using the same estimates as before, I get (cid:12)(cid:12)(cid:12)(cid:12) (cid:18) y f ( x ) − y f ( x ) (cid:19) f ′ ( x t )( x − x ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ x − x ) (cid:12)(cid:12)(cid:12)(cid:12) f ′ ( x t ) f ( x t ) (cid:12)(cid:12)(cid:12)(cid:12) (cid:18) (cid:12)(cid:12)(cid:12)(cid:12) y − y x − x (cid:12)(cid:12)(cid:12)(cid:12) + 2 k (cid:19) ≤ k ( x − x ) f ( x t ) (cid:18) (cid:12)(cid:12)(cid:12)(cid:12) y − y x − x (cid:12)(cid:12)(cid:12)(cid:12) + 2 k (cid:19) . On the other hand, also the second term of the right hand side can be easily estimate: (cid:12)(cid:12)(cid:12)(cid:12)(cid:18) (1 − t ) y f ( x ) + t y f ( x ) (cid:19) f ′′ ( x t )( x − x ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ ( x − x ) . Adding this last two inequalities, and using that f ( x t ) ≤ k I can conclude (cid:12)(cid:12) γ ′′ ( t ) (cid:12)(cid:12) ≤ ( x − x ) (cid:20) kf ( x t ) (cid:18) (cid:12)(cid:12)(cid:12)(cid:12) y − y x − x (cid:12)(cid:12)(cid:12)(cid:12) + 2 k (cid:19)(cid:21) ≤ ( x − x ) kf ( x t ) (cid:18)(cid:12)(cid:12)(cid:12)(cid:12) y − y x − x (cid:12)(cid:12)(cid:12)(cid:12) + 2 (cid:19) , where the last inequality holds if k < . (cid:3) Lemma 3.3.
Given a fixed a constant H , let y : I = [ x , x ] → R + be a C function such that y ′ ( x ) ≥ and y ′′ ( x ) ≤ H kf ( x ) for every x ∈ I . Then, for k small enough, it holds log (cid:18) m (cid:18) x + x , y (cid:18) x + x (cid:19)(cid:19)(cid:19) ≥ log( m ( x , y ( x ))) + log( m ( x , y ( x ))) + K f ( x ) ( x − x ) Proof.
Before going into the proof, I want to point out that the inequality I have to prove isbasically a K -convexity inequality. The strategy of the proof consists in deducing the K -convexityfrom a second derivative estimate. So, start take the first derivative: ∂∂x log( f ( x )) + K (cid:18) y ( x ) f ( x ) (cid:19) ! = f ′ ( x ) f ( x ) + 2 K y ( x ) f ( x ) (cid:18) y ′ ( x ) f ( x ) − y ( x ) f ′ ( x ) f ( x ) (cid:19) Taking another derivative, I obtain ∂ ∂x log( f ( x )) + K (cid:18) y ( x ) f ( x ) (cid:19) ! = f ′′ ( x ) f ( x ) − f ′ ( x ) f ( x ) + 2 K (cid:18) y ′ ( x ) f ( x ) − y ( x ) f ′ ( x ) f ( x ) (cid:19) + 2 K y ( x ) f ( x ) (cid:18) y ′′ ( x ) f ( x ) − y ′ ( x ) f ′ ( x ) f ( x ) − y ( x ) f ′′ ( x ) f ( x ) + 2 y ( x ) f ′ ( x ) f ( x ) (cid:19) . Therefore, neglecting some positive terms, the following estimate holds ∂ ∂x log( f ( x )) + K (cid:18) y ( x ) f ( x ) (cid:19) ! ≥ K y ′ ( x ) f ( x ) − f ′ ( x ) f ( x ) − K (cid:12)(cid:12)(cid:12)(cid:12) y ( x ) y ′ ( x ) f ′ ( x ) f ( x ) (cid:12)(cid:12)(cid:12)(cid:12) − K (cid:12)(cid:12)(cid:12)(cid:12) y ( x ) y ′′ ( x ) f ( x ) (cid:12)(cid:12)(cid:12)(cid:12) − K (cid:12)(cid:12)(cid:12)(cid:12) y ( x ) f ′′ ( x ) f ( x ) (cid:12)(cid:12)(cid:12)(cid:12) . (cid:12)(cid:12)(cid:12) y ( x ) f ( x ) (cid:12)(cid:12)(cid:12) ≤ | f ′ ( x ) | ≤ k and | f ( x ) | ≤ k , I deduce ∂ ∂x log( f ( x )) + K (cid:18) y ( x ) f ( x ) (cid:19) ! ≥ Kf ( x ) (cid:18) y ′ ( x ) − k K − k (cid:12)(cid:12) y ′ ( x ) (cid:12)(cid:12) − | f ( x ) | (cid:12)(cid:12) y ′′ ( x ) (cid:12)(cid:12) − | f ( x ) | (cid:12)(cid:12) f ′′ ( x ) (cid:12)(cid:12) (cid:19) ≥ Kf ( x ) (cid:18) y ′ ( x ) − k − k (cid:12)(cid:12) y ′ ( x ) (cid:12)(cid:12) − kH − k (cid:19) ≥ K f ( x ) , where the last inequality holds for every suitably small k . The thesis follows by making theuniform convexity explicit and noticing that f ( x ) ≥ f ( x ) for every x ∈ I . (cid:3) Performing the same computations of the previous proof, it possible to prove the following corol-lary.
Corollary 3.4.
Given a fixed a constant H , let y : I = [ x , x ] → R + be a C function such that y ′ ( x ) ≥ and y ′′ ( x ) ≤ H kf ( x ) for every x ∈ I . Then, for k small enough, it holds K y (cid:0) x + x (cid:1) f (cid:0) x + x (cid:1) ! ≤ K (cid:18) y ( x ) f ( x ) (cid:19) + K (cid:18) y ( x ) f ( x ) (cid:19) − K f ( x ) ( x − x ) . Lemma 3.5.
Let x ∈ [ − , and δ > such that x − δ, x + δ ∈ [ − , , then (cid:12)(cid:12)(cid:12)(cid:12) f ( x ) f ( x − δ ) + f ( x ) f ( x + δ ) − (cid:12)(cid:12)(cid:12)(cid:12) ≤ [2 k + f ( x )] δ f ( x − δ ) f ( x + δ ) . Proof.
Denote I = R x − δx f ′ ( t ) d t and I = R x + δx f ′ ( t ) d t , then (cid:12)(cid:12)(cid:12)(cid:12) f ( x ) f ( x − δ ) + f ( x ) f ( x + δ ) − (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) f ( x ) f ( x ) + I + f ( x ) f ( x ) + I − (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) − I I − f ( x )( I + I )( f ( x ) + I )( f ( x ) + I ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12) I I ( f ( x ) + I )( f ( x ) + I ) (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12) f ( x )( I + I )( f ( x ) + I )( f ( x ) + I ) (cid:12)(cid:12)(cid:12)(cid:12) . But the following estimates hold | I + I | = (cid:12)(cid:12)(cid:12)(cid:12)Z δ f ′ ( x + t ) − f ′ ( x − δ + t ) d t (cid:12)(cid:12)(cid:12)(cid:12) ≤ Z δ (cid:12)(cid:12) f ′ ( x + t ) − f ′ ( x − δ + t ) (cid:12)(cid:12) d t ≤ δ sup (cid:12)(cid:12) f ′′ (cid:12)(cid:12) ≤ δ and | I | , | I | ≤ kδ. Using this last two estimates I conclude (cid:12)(cid:12)(cid:12)(cid:12) f ( x ) f ( x − δ ) + f ( x ) f ( x + δ ) − (cid:12)(cid:12)(cid:12)(cid:12) ≤ [2 k + f ( x )] δ ( f ( x ) + I )( f ( x ) + I ) (cid:3) How to Prove Convexity of the Entropy
In this section I prove an important result (Proposition 4.2) that will be fundamental in thefollowing, in order to prove the CD condition. This results relies on the possibility to compute thedensity of a pushforward measure, through Jacobi equation. For example, consider two measures µ , µ ∈ P ( R ) which are absolutely continuous with respect to the Lebesgue measure L , withdensity ρ and ρ respectively. Suppose there exists a smooth one-to-one map T : R → R suchthat T µ = µ , then it is well known that ρ ( T ( x, y )) J T ( x, y ) = ρ ( x, y ) , (7)for µ -almost every ( x, y ). As shown in [1] the assumptions on the map T can be relaxed andit is sufficient to require T to be approximately differentiable and injective outside a µ -nullset. However, in this work I am going to deal with transport map which are not necessarilyapproximately differentiable, but have another rigidity property. Therefore, the version of Jacobiequation I will need is the following, which is an easy consequence of Proposition 2.1 in [5]. Proposition 4.1.
Let µ , µ ∈ P ( R ) be absolutely continuous with respect to the Lebesguemeasure L . Assume that there exists a map T = ( T , T ) which is injective outside a µ -nullset, such that T µ = µ . Suppose also that T locally does not depend on the y coordinate andit is increasing in x , while T ( x, y ) is increasing in y for every fixed x . Then the Jacobi equation (7) is satisfied with J T = ∂T ∂x ∂T ∂y . Now that I have shown that Jacobi equation can be adapted to the context of this work, Iam going to explain how it can be useful in proving convexity of the entropy functional. Thefollowing proposition will be an important element in the main proof of this article.
Proposition 4.2.
Let µ , µ ∈ P ( X ) be absolutely continuous measure and let T : X → X be an optimal transport map between µ and µ , in particular T µ = µ . Consider a midpoint µ / of µ and µ , assume that µ / = [ M ◦ (id , T )] µ where the map M : X × X → X is amidpoint selection. Suppose also that the maps T and M ◦ (id , T ) : X → X are injective outsidea set of measure zero and they satisfy the Jacobian equation, with suitable Jacobian functions J T and J M ◦ (id ,T ) . If log (cid:0) m (cid:0) M (( x, y ) , T ( x, y )) (cid:1) J M ◦ (id ,T ) ( x, y ) (cid:1) ≥
12 log ( m ( T ( x, y )) J T ( x, y )) + 12 log( m ( x, y )) for µ almost every ( x, y ) , then Ent( µ / ) ≤
12 Ent( µ ) + 12 Ent( µ ) . (8) Proof.
Suppose µ = ρ m = ˜ ρ L , µ = ρ m = ˜ ρ L and µ / = ρ m = ˜ ρ L . It easy to realizethat, in order to prove (8), it is sufficient to prove thatlog (cid:0) ρ / (cid:0) M (( x, y ) , T ( x, y )) (cid:1)(cid:1) ≤
12 log (cid:0) ρ ( T ( x, y )) (cid:1) + 12 log( ρ ( x, y )) , (9)for µ -almost every ( x, y ). On the other hand, the validity of Jacobi equation ensures that˜ ρ ( T ( x, y )) J T ( x, y ) = ˜ ρ ( x, y ) , for µ -almost every ( x, y ), and thus that ρ ( T ( x, y )) m ( T ( x, y )) J T ( x, y ) = ρ ( x, y ) m ( x, y ) . µ -almost every ( x, y ). Analogously, it is possible to infer that ρ / (cid:0) M (( x, y ) , T ( x, y )) (cid:1) m (cid:0) M (( x, y ) , T ( x, y )) (cid:1) J M ◦ (id ,T ) ( x, y ) = ρ ( x, y ) m ( x, y ) . for µ -almost every ( x, y ). Therefore (9) is equivalent tolog ρ ( x, y ) m ( x, y ) m (cid:0) M (( x, y ) , T ( x, y )) (cid:1) J M ◦ (id ,T ) ( x, y ) ! ≤
12 log (cid:18) ρ ( x, y ) m ( x, y ) m ( T ( x, y )) J T ( x, y ) (cid:19) + 12 log( ρ ( x, y )) . Some easy algebraic computations show that this last equation is equivalent tolog (cid:0) m (cid:0) M (( x, y ) , T ( x, y )) (cid:1) J M ◦ (id ,T ) ( x, y ) (cid:1) ≥
12 log ( m ( T ( x, y )) J T ( x, y )) + 12 log( m ( x, y )) , concluding the proof. (cid:3) Notice that the result of this last proposition gains importance if seen in relation with Proposition1.8. In fact, Proposition 4.2 provides a strategy to prove entropy convexity in a suitable midpoint,which is sufficient to deduce CD condition, according to Proposition 1.8.I conclude the section with a simple corollary of Proposition 4.2, which will be useful in thelast section of this work. This result is a straightforward consequence of the previous proof, andtakes full advantage of the fact that Jacobi equation allows to prove entropy convexity pointwiseas well as globally.
Corollary 4.3.
Under the same assumptions of Proposition 4.2, it holds that
Ent (cid:0) [ M ◦ (id , T )] ( f µ ) (cid:1) ≤
12 Ent( f µ ) + 12 Ent( T ( f µ )) , for every bounded measurable function f : X → R + , with R f d µ = 1 . As previously pointed out, in order to prove CD condition I am going to prove entropy convexityin a suitable midpoint of any pair of absolutely continuous measures. Notice that in an highlybranching metric measure space the choice of a midpoint can be done with great freedom. Thisis because, in general, both the optimal transport map and the geodesic interpolation are notunique, and thus they must be selected in a clever way. In this section, for any pair of absolutelycontinuous measures, I define a suitable midpoint and in the following section I will show that itsatisfies the convexity of the entropy. This midpoint selection is actually quite complicated butit does the job, in particular I believe there is no way to obtain a considerably simpler one.Before going into the details, I introduce the following subsets of R × R that will play animportant role in the definition of the midpoint. Definition 5.1.
Define the sets
V, D, H, H , H , H ⊂ R × R as: V := (cid:8) (( x , y ) , ( x , y )) ∈ R × R : | x − x | < | y − y | (cid:9) ,D := (cid:8) (( x , y ) , ( x , y )) ∈ R × R : | x − x | = | y − y | (cid:9) ,H := (cid:8) (( x , y ) , ( x , y )) ∈ R × R : | x − x | > | y − y | (cid:9) = H ∪ H , where H := (cid:26) (( x , y ) , ( x , y )) ∈ R × R : 12 | x − x | ≥ | y − y | (cid:27) ,H := (cid:26) (( x , y ) , ( x , y )) ∈ R × R : | x − x | > | y − y | > | x − x | (cid:27) . Proposition 5.2.
Given two measures µ , µ ∈ P ( R ) which are absolutely continuous withrespect to the Lebesgue measure L , there exists a measurable optimal transport map T =( T , T ) , injective outside a µ -null set, satisfying T µ = µ , with some nice rigidity properties.In particular the optimal transport plan (id , T ) µ is concentrated in a set Γ ⊂ X × X , such thatfor all ( x, y ) , ( x, y ) , ( x , y ) , ( x , y ) ∈ { ( x, y ) ∈ X : (( x, y ) , T ( x, y )) ∈ Γ } it holds thatif y = y and T ( x, y ) = T ( x, y ) , then T ( x, y ) − T ( x, y ) y − y ≥ and if x = x and T ( x , y ) = T ( x , y ) , then T ( x , y ) − T ( x , y ) x − x ≥ . Moreover for µ -almost every ( x, y ) I have that T is locally constant in y, if (( x, y ) , T ( x, y )) ∈ H and T is locally constant in x, if (( x, y ) , T ( x, y )) ∈ V .
Combining this two properties with some monotonicity properties one can deduce that the func-tion T ( x, y ) is increasing in x for every fixed y and the function T ( x, y ) is increasing in y forevery fixed x , as a consequence for µ -almost every ( x, y ) it holds ∂T ∂x ≥ and ∂T ∂y ≥ , if (( x, y ) , T ( x, y )) ∈ H ∪ V. Finally let me point out that, since (id , T ) µ ∈ OptPlans( µ , µ ) the usual cyclical monotonicityholds, thus I can assume that for every ( z , w ) , ( z , w ) ∈ Γ it holds d ∞ ( z , w ) + d ∞ ( z , w ) ≤ d ∞ ( z , w ) + d ∞ ( z , w ) . Now fix two measures µ , µ ∈ P ( X ) absolutely continuous with respect to the referencemeasure m , and thus also with respect to the Lebesgue measure L . Call T the optimal transportmap between µ and µ , that satisfies the requirements of Proposition 5.2. Moreover denote byΓ the full (id , T ) µ -measure set with all the monotonicity property stated in Proposition 5.2.In order to identify a midpoint of µ and µ , I need to choose a proper midpoint interpolation,that is a measurable map M : X × X → X such that d ∞ ( M ( z, w ) , z ) = d ∞ ( M ( z, w ) , w ) = 12 d ∞ ( z, w ) for every ( z, w ) ∈ X × X, (10)the desired midpoint will be M (cid:0) (id , T ) µ (cid:1) = [ M ◦ (id , T )] µ .Let me now define the midpoint interpolation map M that I will use for the proof of the maintheorem. This definition is actually quite involved, in fact the map M is defined in different wayson the sets V , D , H and H . In particular the precise definition is the following: • If (cid:0) ( x , y ) , ( x , y ) (cid:1) ∈ V ∪ DM (cid:0) ( x , y ) , ( x , y ) (cid:1) := (cid:18) x + x , y + y (cid:19) If (cid:0) ( x , y ) , ( x , y ) (cid:1) ∈ H , M (cid:0) ( x , y ) , ( x , y ) (cid:1) = (cid:18) x + x , (cid:18) y f ( x ) + y f ( x ) (cid:19) f (cid:18) x + x (cid:19)(cid:19) . • If (cid:0) ( x , y ) , ( x , y ) (cid:1) ∈ H , with x < x and y < y , introduce the quantity˜ y ( x , x , y ) = 12 (cid:18) y f ( x ) + y + x − x f ( x ) (cid:19) f (cid:18) x + x (cid:19) − y , and consequently define M (cid:0) ( x , y ) , ( x , y ) (cid:1) = (cid:18) x + x , y + ˜ y ( x , x , y ) + (cid:18) x − x − ˜ y ( x , x , y ) (cid:19) (cid:18) y − y x − x − (cid:19)(cid:19) . In the other cases M can be defined analogously, but I prefer not to explicitly do it, in orderto avoid unnecessary complications in this definition and in the following. In particular everyproof from now on will be done using only the definition above, implying it can be easilyadapted to the other cases.With such a complex definition is not completely obvious that the map M satisfies condition(10), but this can be easily proven. Proposition 5.3.
The map M actually defines a midpoint interpolation. Proof.
This is completely obvious if (cid:0) ( x , y ) , ( x , y ) (cid:1) ∈ V ∪ D . While when (cid:0) ( x , y ) , ( x , y ) (cid:1) ∈ H the statement is an easy consequence of Lemma 3.2, provided that k is sufficiently small, andof the fact that 0 ≤ (cid:18) y f ( x ) + y f ( x ) (cid:19) f (cid:18) x + x (cid:19) ≤ f (cid:18) x + x (cid:19) . On the other hand if (cid:0) ( x , y ) , ( x , y ) (cid:1) ∈ H it is sufficient to notice that the point(¯ x, ¯ y ) := (cid:18) x + x , y + ˜ y ( x , x , y ) + (cid:18) x − x − ˜ y ( x , x , y ) (cid:19) (cid:18) y − y x − x − (cid:19)(cid:19) lies on a convex combination of a 45 degree line and a curve γ : [ x , x ] → X , with12 − k ≤ γ ′ ( x ) ≤
12 + 3 k for every x ∈ [ x , x ] , according to Lemma 3.2. As a consequence, for a suitably small k , this ensures both that (¯ x, ¯ y ) ∈ X and that (¯ x, ¯ y ) is a d ∞ -midpoint of ( x , y ) and ( x , y ). (cid:3) In order to efficiently apply Proposition 4.2, I need to compute the Jacobian of the map M ◦ (id , T ). Observe that the way M is defined, combined with the properties of T , ensuresthat M ◦ (id , T ) satisfies all the rigidity assumptions of Proposition 4.1. Therefore, proving thefollowing result, will allow to use Proposition 4.1 to compute J M ◦ (id ,T ) . Proposition 5.4.
The map M ◦ (id , T ) is injective outside a µ -null set. Proof.
First of all notice that it is sufficient to prove the injectivity of M on Γ, because Γ has full(id , T ) µ -measure. Thus suppose by contradiction that there exist(( x , y ) , ( x ′ , y ′ )) = (( x , y ) , ( x ′ , y ′ )) ∈ Γ14uch that M (( x , y ) , ( x ′ , y ′ )) = M (( x , y ) , ( x ′ , y ′ )) . Following the proof of Lemma 3.7 in [5], I can limit myself to consider the case when(( x , y ) , ( x ′ , y ′ )) = (( x , y ) , ( x ′ , y ′ )) ∈ Γ ∩ H. In this case, one can easily realize that the cyclical monotonicity properties of Γ imply that x = x and x ′ = x ′ . So, if (( x , y ) , ( x ′ , y ′ )) , (( x , y ) , ( x ′ , y ′ )) ∈ Γ ∩ H , thesis simply follows from thedefinition of M and from the monotonicity of T . While if (( x , y ) , ( x ′ , y ′ )) , (( x , y ) , ( x ′ , y ′ )) ∈ Γ ∩ H the statement is a consequence of the monotonicity property of T associated with thefact that the quantity y + ˜ y ( x, x ′ , y ) + (cid:18) x ′ − x − ˜ y ( x, x ′ , y ) (cid:19) (cid:18) y ′ − yx ′ − x − (cid:19) is locally increasing in y and y ′ , when (( x, y ) , ( x ′ , y ′ )) ∈ H (with x < x ′ and y < y ′ ). The firstmonotonicity is not straightforward, therefore I am going to prove it. First of all notice that,according to Lemma 3.5 and since f ≤ k , it holds that2 (cid:12)(cid:12)(cid:12)(cid:12) ∂∂y ˜ y ( x, x ′ , y ) (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) f (cid:0) x + x ′ (cid:1) f ( x ) + f (cid:0) x + x ′ (cid:1) f ( x ′ ) − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ [2 k + 3 k ] (cid:0) x ′ − x (cid:1) f ( x ) f ( x ′ ) . Moreover, the geometry of the set X allows to deduce that x ′ − x ≤ y + x ′ − x ≤ f ( x ′ ) ≤ f ( x ) + k ( x ′ − x ) , and consequently (cid:0) − k (cid:1) x ′ − x ≤ f ( x ) ≤ f ( x ′ ). On the other hand, observe that Lemma 3.2guarantees that (cid:18) x ′ − x − ˜ y ( x, x ′ , y ) (cid:19) ≤ (cid:18)
12 + 3 k (cid:19) x ′ − x ∂∂y (cid:20) y + ˜ y ( x, x ′ , y ) + (cid:18) x ′ − x − ˜ y ( x, x ′ , y ) (cid:19) (cid:18) y ′ − yx ′ − x − (cid:19) (cid:21) ≥ − k − (cid:12)(cid:12)(cid:12)(cid:12) ∂∂y ˜ y ( x, x ′ , y ) (cid:12)(cid:12)(cid:12)(cid:12) (cid:18) − y ′ − yx ′ − x (cid:19) ≥ − k − [2 k + 3 k ](1 − k ) > , for k sufficiently small. The case when (( x , y ) , ( x ′ , y ′ )) ∈ Γ ∩ H , (( x , y ) and ( x ′ , y ′ )) ∈ Γ ∩ H can be treated analogously. (cid:3) In the previous sections I have introduced all I need to prove that the metric measure space( X, d ∞ , m ) satisfies the CD (0 , ∞ ) condition. Let me now go into the details of the proof. Theorem 6.1.
For suitable k and K , the metric measure space ( X k,ε , d ∞ , m k,K,ε ) is a CD (0 , ∞ ) space, for every < ε < k . roof. Let µ , µ ∈ P ( X ) be absolutely continuous with respect to the reference measure m ,then, according to Proposition 1.8, it is sufficient to prove thatEnt (cid:0) [ M ◦ (id , T )] µ (cid:1) ≤
12 Ent( µ ) + 12 Ent( µ ) . Given Proposition 4.2, it is enough check the validity oflog (cid:0) m (cid:0) M (( x, y ) , T ( x, y )) (cid:1) J M ◦ (id ,T ) ( x, y ) (cid:1) ≥
12 log ( m ( T ( x, y )) J T ( x, y )) + 12 log( m ( x, y )) (11)for µ -almost every ( x, y ). For µ -almost every ( x, y ) ∈ V ∪ D this can be done following [5].Thus I will treat the other cases and applying Lemma 3.3.Notice that, for µ -almost every ( x, y ) such that (( x, y ) , T ( x, y )) ∈ H ∩ Γ, I have M ◦ (id , T )( x, y ) = (cid:18) x + T , (cid:18) yf ( x ) + T f ( T ) (cid:19) f (cid:18) x + T (cid:19)(cid:19) . Then, according to what I did in previous sections, it is possible to apply Proposition 4.1 anddeduce that J M ◦ (id ,T ) ( x, y ) = 12 (cid:18) ∂T ∂x (cid:19) (cid:18) f ( x ) + ∂T ∂y f ( T ) (cid:19) f (cid:18) x + T (cid:19) , for µ -almost every ( x, y ) such that (( x, y ) , T ( x, y )) ∈ H ∩ Γ. Furthermore it holds that m (cid:0) M (( x, y ) , T ( x, y )) (cid:1) = f (cid:18) x + T (cid:19) − exp − K (cid:18) yf ( x ) + T f ( T ) (cid:19) ! , thus, putting together this last two relations, I obtainlog (cid:0) m (cid:0) M (( x, y ) , T ( x, y )) (cid:1) J M ◦ (id ,T ) ( x, y ) (cid:1) = log (cid:18) ∂T ∂x (cid:19) (cid:18) f ( x ) + ∂T ∂y f ( T ) (cid:19) exp − K (cid:18) yf ( x ) + T f ( T ) (cid:19) !! = log (cid:18) (cid:18) ∂T ∂x (cid:19)(cid:19) + log (cid:18) f ( x ) + ∂T ∂y f ( T ) (cid:19)! − K (cid:18) (cid:18) yf ( x ) + T f ( T ) (cid:19)(cid:19) On the other hand it holdslog( m ( x, y )) = log f ( x ) exp − K (cid:18) yf ( x ) (cid:19) !! = log(1) + log (cid:18) f ( x ) (cid:19) − K (cid:18) yf ( x ) (cid:19) and, applying once again Proposition 4.1, this time to the map T , alsolog ( m ( T ( x, y )) J T ( x, y )) = log ∂T ∂x ∂T ∂y f ( T ) exp − K (cid:18) T f ( T ) (cid:19) !! = log (cid:18) ∂T ∂x (cid:19) + log ∂T ∂y f ( T ) ! − K (cid:18) T f ( T ) (cid:19) , for µ -almost every ( x, y ) such that (( x, y ) , T ( x, y )) ∈ H ∩ Γ. Putting together this last threeequations, inequality (11) follows from the concavity of the functions log and − Kx .16assing now to the last case, for µ -almost every ( x, y ) such that (( x, y ) , T ( x, y )) ∈ H ∩ Γ (with x < T ( x, y ) and y < T ( x, y )) I have( S , S )( x, y ) := M ◦ (id , T )( x, y )= (cid:18) x + T , y + ˜ y ( x, T , y ) + (cid:18) T − x − ˜ y ( x, T , y ) (cid:19) (cid:18) T − yT − x − (cid:19)(cid:19) . Reasoning as before, Proposition 4.1 ensures that, for µ -almost every ( x, y ) such that (( x, y ) , T ( x, y )) ∈ H ∩ Γ (with x < T ( x, y ) and y < T ( x, y )), J M ◦ (id ,T ) ( x, y ) = ∂S ∂x ∂S ∂y , and in particular it holds that ∂S ∂x = 12 (cid:18) ∂T ∂x (cid:19) , and ∂S ∂y = 1 + ∂∂y ˜ y ( x, T , y ) (cid:18) − T − yT − x (cid:19) + 2 (cid:18) T − x − ˜ y ( x, T , y ) (cid:19) ∂T ∂y − T − x = 1 + ∂∂y ˜ y ( x, T , y ) (cid:18) − T − yT − x (cid:19) + (cid:18) ∂T ∂y − (cid:19) − ˜ y ( x, T , y ) T − x ! = 1 + ∂∂y ˜ y ( x, T , y ) (cid:18) − T − yT − x (cid:19) + (cid:18) ∂T ∂y − (cid:19) − ˜ y ( x, T , y ) − T − x T − x ! . I can now consider the explicit value of ˜ y ( x, T , y ) and notice that˜ y ( x, T , y ) − x + T "(cid:18) yf ( x ) + y + T − x f ( T ) (cid:19) f (cid:18) T + x (cid:19) − y − T − x = 12 " y f (cid:0) T + x (cid:1) f ( x ) + f (cid:0) T + x (cid:1) f ( T ) − ! + T − x f (cid:0) T + x (cid:1) f ( T ) − ! . Moreover, I can easily obtain that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) f (cid:0) T + x (cid:1) f ( T ) − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) f (cid:0) T + x (cid:1) − f ( T ) f ( T ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ sup f ′ · T − x f ( T ) = k · T − x f ( T ) , thus applying Lemma 3.5 and noticing that yf ( x ) ≤
1, I can conclude that (cid:12)(cid:12)(cid:12)(cid:12) ˜ y ( x, T , y ) − T − x (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:2) k + k + f (cid:0) T + x (cid:1)(cid:3) (cid:0) T − x (cid:1) f ( T ) ≤ [2 k + 4 k ] (cid:0) T − x (cid:1) f ( T )where in the last inequality I used that f ( x ) ≤ k for every x ∈ [ − , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˜ y ( x, T , y ) − T − x T − x (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ [2 k + 4 k ] T − x f ( T ) ≤ T − x f ( T ) ≤ T − x f ( T ) , (12)for a sufficiently small k . Now suppose that ∂∂y ˜ y ( x, T , y ) = 12 f (cid:0) x + T (cid:1) f ( x ) + 12 f (cid:0) x + T (cid:1) f ( T ) − > , X ensures that T − x ≤ f ( T ), it is possible toapply Lemma 3.1 and obtain thatlog (cid:18) ∂S ∂y (cid:19) ≥ log (cid:18) ∂T ∂y − (cid:19) − ˜ y ( x, T , y ) − T − x T − x !! ≥ log(1) + log (cid:18) ∂T ∂y (cid:19) − C (cid:0) T − x (cid:1) f ( T ) . (13)On the other hand, it is easy to realize that the point M ◦ (id , T )( x, y ) lies on a curve, which isa convex combination of a 45 degree line and of the curve t (cid:18) (1 − t ) yf ( x ) + t y + T − x f ( T ) (cid:19) f ((1 − t ) x + tT ) . Therefore, up to take a suitably small k , Lemma 3.2 allows to apply Lemma 3.3, and obtainlog ( m ( M ◦ (id , T )( x, y ))) ≥ log( m ( x, y )) + log( m ( T ( x, y ))) + K f ( T ) ( T − x ) . (14)Inequality (11) then follows as before, putting together (13) and (14) and taking K sufficientlylarge.Suppose instead that ∂∂y ˜ y ( x, T , y ) = 12 f (cid:0) x + T (cid:1) f ( x ) + 12 f (cid:0) x + T (cid:1) f ( T ) − < , then notice thatlog (cid:18) f (cid:18) T + x (cid:19)(cid:19) − (cid:0) log( f ( x )) + log( f ( T )) (cid:1) = 12 log f (cid:0) T + x (cid:1) f ( x ) · f (cid:0) T + x (cid:1) f ( T ) ! ≤ log f (cid:0) T + x (cid:1) f ( x ) + 12 f (cid:0) T + x (cid:1) f ( T ) ! = log (cid:18) ∂∂y ˜ y ( x, T , y ) (cid:19) (15)Moreover, according to the estimates done in the proof of Proposition 5.4 and to (12), it is easyto realize that, for k small enough − ˜ y ( x, T , y ) − T − x T − x ! (cid:18) ∂∂y ˜ y ( x, T , y ) (cid:19) − = 12 + ˜ δ for some ˜ δ such that (cid:12)(cid:12) ˜ δ (cid:12)(cid:12) ≤ T − x f ( T ) . Consequently I can infer thatlog (cid:18) ∂S ∂y (cid:19) ≥ log ∂∂y ˜ y ( x, T , y ) + (cid:18) ∂T ∂y − (cid:19) − ˜ y ( x, T , y ) − T − x T − x !! = log (cid:18) ∂∂y ˜ y ( x, T , y ) (cid:19) + log (cid:18) (cid:18) ∂T ∂y − (cid:19) (cid:18)
12 + ˜ δ (cid:19)(cid:19) ≥ log (cid:18) ∂∂y ˜ y ( x, T , y ) (cid:19) + log(1) + log (cid:18) ∂T ∂y (cid:19) − C (cid:0) T − x (cid:1) f ( T ) (16)where the last passage follows from Lemma 3.1. Finally it is possible to prove (11), puttingtogether (15) with (16), applying Corollary 3.4 and taking K sufficiently large. (cid:3)
18s I did in section 4, I exploit the local nature of Jacobi equation to improve the last result.The following result is an easy consequence of Theorem 6.1 and Corollary 4.3, and it will be usefulin the end of this work.
Corollary 6.2.
Given two absolutely continuous measures µ , µ ∈ P ( X ) , assume that thereexists a map T such that T µ = µ , satisfying all the properties of Proposition 5.2. Then, calling M the midpoint selection presented in section 5, it holds Ent (cid:0) [ M ◦ (id , T )] ( f µ ) (cid:1) ≤
12 Ent( f µ ) + 12 Ent( T ( f µ )) , for every bounded measurable function f : X → R + with R f d µ = 1 . In this last section I conclude all the work done up to now. In particular I am going to show whythis example is relevant, asking also to some open question related to strict curvature dimensionbounds. First of all let me prove the most important result, which had already been anticipatedin previous sections.
Theorem 7.1.
For suitable k and K the metric measure space ( X k, , d ∞ , m k,K, ) is a CD (0 , ∞ ) space. Proof.
I am going to prove that, for every sequence of positive real numbers ( ε n ) n ∈ N convergingto zero, the sequence of metric measure spaces ( X k,ε n , d ∞ , m k,K,ε n ) measured Gromov Hausdorffconverges to ( X k, , d ∞ , m k,K, ). According to Theorem 1.15 and Theorem 6.1 this is sufficient toconclude the proof, up to choose suitable k and K .Define the function f n : X k,ε n → X k, as f n ( x, y ) = (cid:18) x, y · f k, ( x ) f k,ε n ( x ) (cid:19) , it is immediate to notice that its image is actually X k, . Moreover its is easy to prove that( f n ) m k,K,ε n = m k,K, and (cid:12)(cid:12) d ∞ (cid:0) f n ( x , y ) , f n ( x , y ) (cid:1) − d ∞ (cid:0) ( x , y ) , ( x , y ) (cid:1)(cid:12)(cid:12) ≤ ε n , and this shows the desired measured Gromov-Hausdorff convergence. (cid:3) First of all notice that the space ( X k, , d ∞ , m k,K, ) has different topological dimensions atdifferent regions of the space. In particular, this shows the non-constancy of topological dimensionalso for CD spaces, extending one of the results by Ketterer and Rajala [3]. Furthermore, the space( X k, , d ∞ , m k,K, ) is not a very strict CD ( K, ∞ ) space for every K ∈ R , in fact it is not weaklyessentially non-branching (see Theorem 1.12). In order to see this, it is sufficient to consideran absolutely continuous measure µ supported on L and an absolutely continuous measure µ supported on C , and subsequently notice that every η ∈ OptGeo( µ , µ ) is supported in branchinggeodesics. It is then possible to conclude that the weak CD condition is not sufficient to ensureany type of non-branching condition. Observe also that every η ∈ OptGeo( µ , µ ) is not inducedby a map, consequently the (weak) CD condition is not sufficient to ensure the existence of anoptimal transport map, between two absolutely continuous marginals. Finally notice that thespace ( X k, , d ∞ , m k,K, ) is an example of (weak) CD space which is not a very strict CD space,and this shows that this two notions of curvature dimension bounds are actually different.For the last part of this section I need to introduce another type of curvature bounds, calledstrict CD condition, which is stronger than the weak CD condition, but is weaker than the verystrict one. 19 efinition 7.2. A metric measure space ( X, d , m ) is called a strict CD ( K, ∞ ) space if for ev-ery absolutely continuous measures µ , µ ∈ P ( X ) there exists an optimal geodesic plan η ∈ OptGeo( µ , µ ), so that the entropy functional Ent satisfies the K-convexity inequality along f η for every bounded measurable function f : Geo( X ) → R + with R f d η = 1.I am now going to prove that, for suitable constants, the spaces ( X k,ε , d ∞ , m k,K,ε ) with 0 < ε The strict CD condition is not stable under measured Gromov Hausdorffconvergence. Before going into the details of the proofs, I want to make some clarifications. The fact thatthe spaces ( X k,ε , d ∞ , m k,K,ε ) are strict CD is a consequence of Corollary 6.2 and of an iterationargument. On the other hand the space ( X k, , d ∞ , m k,K, ) cannot be a strict CD space, becauseof its particular topological structure I have already highlighted. Proposition 7.4. For suitable k and K and every < ε < k the metric measure space ( X k,ε , d ∞ , m k,K,ε ) is a strict CD (0 , ∞ ) space. Proof. For every n ∈ N I am going to define a measurable map G n : X → Geo( X ) by induction.In particular I introduce G : X → Geo( X ) as any measurable map such that ( e , e ) ◦ G ( x ) =(id , T ) µ -almost everywhere, consequently ( G ) µ ∈ OptGeo( µ , µ ). Given G n : X → Geo( X ), define G n +1 : X → Geo( X ) by imposing that:1. e r ◦ G n +1 = e r ◦ G n for every r ∈ (cid:8) k n , k = 0 , . . . , n (cid:9) e k +12 n +1 ◦ G n +1 = M (cid:0) e k n ◦ G n , e k +12 n ◦ G n (cid:1) µ -almost everywhere, where M is the midpointmap defined as in Section 5.Notice that, if the optimal transport map that induces (cid:0) e k n ◦ G n , e k +12 n ◦ G n (cid:1) µ satisfies all theproperties of Proposition 5.2, then also the maps that induce (cid:0) e k n +1 ◦ G n +1 , e k +12 n +1 ◦ G n +1 (cid:1) µ and (cid:0) e k +12 n +1 ◦ G n +1 , e k +22 n +1 ◦ G n +1 (cid:1) µ satisfy all the properties of Proposition 5.2. The reader caneasily realize that this is a quite straightforward consequence of the definition of map M and ofits properties highlighted in Section 6. This observation shows that the inductive procedure Ihave introduced can be done in accordance with the previous section, moreover it is possible toapply Corollary 6.2 and obtain thatEnt (cid:0) ( e s ) [( G n +1 ) ( f µ )] (cid:1) ≤ 12 Ent (cid:0) ( e r ) [( G n +1 ) ( f µ )] (cid:1) + 12 Ent (cid:0) ( e t ) [( G n +1 ) ( f µ )] (cid:1) − K W (cid:0) ( e r ) [( G n +1 ) ( f µ )] , ( e t ) [( G n +1 ) ( f µ )] (cid:1) , (17)where r = k n +1 , s = k +12 n +1 , t = k +22 n +1 and f is any bounded measurable function with R f d µ = 1.Notice that, in order to infer (17), I also used that the map e r ◦ G n is injective outside a µ -nullset, as a consequence of Proposition 5.4.An inductive argument allows to conclude that for every n ∈ N it holdsEnt (cid:0) ( e r ) [( G n ) ( f µ )] (cid:1) ≤ (1 − r ) Ent( f µ ) + r Ent( T ( f µ )) − K r (1 − r ) W ( f µ , T ( f µ )) , for every r ∈ (cid:8) k n , k = 0 , . . . , n (cid:9) , and every bounded measurable function f with R f d µ = 1.In fact, this is completely obvious for n = 0, and assuming it true for an n it is possible to easily20 µ µ s Figure 3: A representation of the geodesic ( µ t ) t ∈ [0 , .deduce it for n + 1, using (17).It is now easy to notice that the first property in the definition of G n +1 given G n , ensures theexistence of a measurable map G : X → Geo( X ), such that G n → G uniformly. Furthermore itis obvious that ( e r ) [( G n ) ( f µ )] = ( e r ) [ G ( f µ )]for every r ∈ (cid:8) k n , k = 0 , . . . , n (cid:9) , and every bounded measurable function f with R f d µ = 1.Consequently it holds thatEnt (cid:0) ( e r ) [ G ( f µ )] (cid:1) ≤ (1 − r ) Ent(( f µ )) + r Ent( T ( f µ )) − K r (1 − r ) W ( f µ , T ( f µ )) , for every dyadic time r and every suitable function f . Then the lower semicontinuity of Entallows to infer that the K -convexity inequality of the entropy is satisfied along G ( f µ ) for everysuitable function f . Finally I can conclude by observing that every optimal geodesic plan of thetype F · G µ , for a measurable function F : Geo( X ) → R + with R F d( G µ ) = 1, can bewritten as G ( f µ ) for a suitable measurable f with R f d µ = 1, since G is clearly injective. (cid:3) Proposition 7.5. For every k and K the metric measure space ( X k, , d ∞ , m k,K, ) is not a strict CD (0 , ∞ ) space. Proof. In this proof I denote with m the measure m k,K, , in order to simplify the notation.For every t ∈ [0 , 1] define the measure µ t = 1 m (cid:0) [ − + t, − + t ] × R (cid:1) · m | [ − + t, − + t ] × R = 4 C K · m | [ − + t, − + t ] × R , see Figure 3 in order to visualize it. It is easy to realize that ( µ t ) t ∈ [0 , is the unique geodesic con-necting µ and µ , along which the entropy functional is convex. Moreover let η ∈ P (cid:0) C ([0 , , X (cid:1) such that ( e t ) η = µ t for every t ∈ [0 , F η , for a suitable bounded measurable function F : Geo( X ) → R + with R F d η = 1.Before going on, let me point out that every Wasserstein geodesic in P (cid:0) C ([0 , , X (cid:1) which con-nects a measure on L to a measure on C (and thus η in particular), consists only of “horizontal”transports. Therefore the only useful coordinate, in order to evaluate the distance d ∞ , will be the x coordinate. As a consequence every such optimal geodesic plan (and η in particular) will onlydepend on the x coordinate. Some of the considerations I will do in this proof actually followsfrom this observation.Now, define the set A := { ( x, y ) ∈ R : f k, ( x ) > ≤ y ≤ f k, ( x ) } ⊂ X k, , C ′ K = Z e − Ky d y. Then fix a time ¯ t such that µ ¯ t is concentrated in C , consider the map ˜ F : Geo( X ) → R + definedas ˜ F := C K C ′ K · χ A ◦ e ¯ t and call ˜ µ = ( e ) ( ˜ F η ). Notice that R ˜ F d η = 1, thus ˜ µ is a probabilitymeasure and it is absolutely continuous with respect to m , with density ˜ ρ bounded above by C ′ K , as a consequence m ( { ˜ ρ > } ) ≥ C ′ K . Now, suppose that m ( { ˜ ρ > } ) = C ′ K , then ˜ ρ ≡ C ′ K on { ˜ ρ > } and therefore Ent(˜ µ ) = log (cid:18) C ′ K (cid:19) = Ent (cid:0) ( e ¯ t ) ( ˜ F η ) (cid:1) . On the other hand Ent (cid:0) ( e ) ( ˜ F η ) (cid:1) = Ent( µ ) = log (cid:18) C K (cid:19) < log (cid:18) C ′ K (cid:19) , and consequently the entropy functional is not convex along ˜ F η .Otherwise, suppose that m ( { ˜ ρ > } ) > C ′ K , call S := { ˜ ρ > } and define the set S x := { ( x ′ , y ′ ) ∈ S : x ′ = x } , for every x ∈ [ − , m := ( p ) m and denote by ( m x ) x ∈ [ − , ⊂ P ( R )the disintegration of m with respect to the projection map p . Notice that, since η depends onlyon the x coordinate, then ( p ) ˜ µ = C K · m | [ , ]. Moreover, since the density ˜ ρ is boundedabove by C ′ K , it holds that m x ( S x ) ≥ C ′ K C K , for m -almost every x ∈ (cid:2) , (cid:3) . Furthermore, theassumption on S , that is m ( S ) > C ′ K , ensures that m x ( S x ) > C ′ K C K for a m | [ , ]-positive set of x ,therefore Z log ( m x ( S x )) d m ( x ) > C K (cid:18) C ′ K C K (cid:19) . (18)On the other hand, for every positive constant c > S c := { ˜ ρ > c } and call S cx := { ( x ′ , y ′ ) ∈ S c : x ′ = x } for every x ∈ [ − , c , since ˜ ρ is bounded and ( p ) ˜ µ = C K · m | [ , ], the quantity m x ( S x ) is uniformlybounded from below for m -almost every x ∈ (cid:2) , (cid:3) . Consequently, it is possible to apply themonotone convergence theorem and deduce that there exists a constant ¯ c > Z log (cid:0) m x ( S ¯ cx ) (cid:1) d m ( x ) > C K (cid:18) C ′ K C K (cid:19) . Now, define the measurable map F : Geo( X ) → R + F := 4 C K ˜ F · (cid:18) m x ( S ¯ cx ) χ S ¯ c ( x, y )˜ ρ ( x, y ) ◦ e (cid:19) . I have already noticed that the quantity m x ( S ¯ cx ) is uniformly bounded from below for m -almostevery x ∈ (cid:2) , (cid:3) , thus F is well defined and bounded, moreover it holds that Z F d η = Z C K · (cid:18) m x ( S ¯ cx ) χ S ¯ c ( x, y )˜ ρ ( x, y ) ◦ e (cid:19) d ˜ F η = 4 C K Z m x ( S ¯ cx ) χ S ¯ c ( x, y )˜ ρ ( x, y ) d˜ µ ( x, y )= 4 C K Z χ S ¯ c ( x, y ) m x ( S ¯ cx ) d m ( x, y ) = 4 C K Z Z χ S ¯ c ( x, y ) m x ( S ¯ cx ) d m x ( y ) d m ( x ) = 4 C K Z d m ( x ) = 1 . 22n particular, observe that a computation similar to this last one shows that( e ) ( F η ) = 4 C K χ S ¯ c ( x, y ) m x ( S ¯ cx ) · m , thus it is possible to estimate its entropy:Ent (cid:0) ( e ) ( F η ) (cid:1) = Z C K χ S ¯ c ( x, y ) m x ( S ¯ cx ) log (cid:18) C K χ S ¯ c ( x, y ) m x ( S ¯ cx ) (cid:19) d m = Z Z C K χ S ¯ c ( x, y ) m x ( S ¯ cx ) log (cid:18) C K χ S ¯ c ( x, y ) m x ( S ¯ cx ) (cid:19) d m x ( y ) d m ( x )= Z Z S ¯ cx C K m x ( S ¯ cx ) log (cid:18) C K m x ( S ¯ cx ) (cid:19) d m x ( y ) d m ( x )= 4 C K Z log (cid:18) C K m x ( S ¯ cx ) (cid:19) d m ( x )= log (cid:18) C K (cid:19) + 4 C K Z − log (cid:0) m x ( S ¯ cx ) (cid:1) d m ( x ) < log (cid:18) C ′ K (cid:19) , where the last inequality follows from (18). On the other hand ( e ¯ t ) ( F η ) ≪ ( e ¯ t ) ( ˜ F η ) andconsequently Jensen’s inequality ensures thatEnt (cid:0) ( e ¯ t ) ( F η ) (cid:1) ≥ log (cid:18) C ′ K (cid:19) , Furthermore, it is easy to realize that( p ) (cid:2) ( e ) ( F η ) (cid:3) = 4 C K · m | [ , ]and thus, as before, I haveEnt (cid:0) ( e ) ( F η ) (cid:1) = Ent( µ ) = log (cid:18) C K (cid:19) < log (cid:18) C ′ K (cid:19) . Putting together this last three inequalities it is easy to realize that the entropy functional is notconvex along F η . (cid:3) Notice that this last result shows in particular that the strict CD condition and the weak one aretwo actually different notions. Moreover, the combination of Proposition 7.4 and 7.5 obviouslyyields Proposition 7.3, according to the proof of Theorem 7.1. On the other hand, observe thatthis work does not allow to disprove the stability of the very strict CD condition. 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