FOR METRIC MEASURE SPACES
ANCA-IULIANA BONCIOCAT
This is a survey of recent developments in the area of lower Ricci curvature bounds for both geodesic metric measure spaces, and discrete metric measure spaces, from a unitary perspective, based on optimal mass transportation.
AMS 2010 Subject Classification: 49Q20, 53C23, 52C99.
Key words: optimal transport, Ricci curvature, graphs, concentration of measure, curvature-dimension condition.
1. INTRODUCTION
The exploration of Riemannian geometry through mass transportation has raised the interest of many mathematicians during the last decade, in works by F. Otto, C. Villani, D. Cordero-Erausquin, R. McCann, M. Schmuck- enschl¨ager, K.T. Sturm and others. In the paper [18], lower Ricci curvature bounds for Riemannian manifolds have been characterized in terms of dis- placement convexity of the entropy functional on the Wasserstein space.
Moreover, discrete mathematics has become popular in the recent de- cades because of its applications to computer science. Triangulations of mani- folds and discretizations of continuous spaces are very useful tools in digital geometry or computational geometry. The digital geometry deals with two main problems, inverse to each other: on one hand constructing digitized representations of objects, with a special emphasize on efficiency and preci- sion, and on the other hand reconstructing “real” objects or their properties (length, area, volume, curvature, surface area) from digital images. Such a study requires of course a better understanding of the geometrical aspects of discrete spaces.
As a first step, geodesic metric spaces are very natural generalizations of manifolds. There are many recent developments in studying the geometry of such spaces. Even from the fifties a notion of lower curvature bounds for metric spaces was introduced by Alexandrov in [1], in terms of comparison properties for geodesic triangles. This notion gives the usual sectional curvature bounds
MATH. REPORTS14(64),3 (2012), 253–278
when applied to Riemannian manifolds and it is stable under the Gromov- Hausdorff convergence, introduced in [10].
More recently, a generalized notion of Ricci curvature bounds for metric measure spaces (M,d, m) has been introduced and studied by K.T. Sturm in [19]; a closely related theory has been developed independently by J. Lott and C. Villani in [14]. The approach presented in [19] is based on convexity properties of the relative entropy Ent(·|m) as a function on theL2-Wasserstein space of probability measures on the metric space (M,d). This lower curvature bound is stable under an appropriate notion ofD-convergence of metric mea- sure spaces. The second paper [20] has treated the “finite dimensional” case, namely metric measure spaces that satisfy the so-called curvature-dimension conditionCD(K, N), whereKplays the role of the lower curvature bound and N the one of the upper dimension bound. The condition CD(K, N) repre- sents the geometric counterpart of the analytic curvature-dimension condition introduced by D. Bakry and M. ´Emery in [3].
These generalizations require the Wasserstein space of probability mea- sures (and thus in turn the underlying space) to be a geodesic space. There- fore, in the original form they cannot apply to discrete spaces. Moreover, if we consider a graph, more precisely the union of the edges of a graph, as a metric space it has no lower curvature bound in the sense of [19], since the vertices are branch points of geodesics which does not allow the K-convexity of the entropy.
The coarse geometry studies the “large scale” properties of spaces (see for instance [17] for an introduction). In various contexts, one notices that the relevant geometric properties of metric spaces are the coarse ones. A discrete space can get a geometric shape when we move the observation point far away from it. It is the point of view that led M. Gromov to his notion of hyperbolic group, which is a group “coarsely negatively curved” (in a certain combinatorial sense).
The paper [5] develops a notion of lower rough curvature bounds for discrete spaces, based on the concept of optimal mass transportation. These rough curvature bounds depend on a real parameter h >0, which should be considered as a natural length scale of the underlying discrete space or as the scale on which we have to look at the space. For a metric graph, for instance, this parameter equals the maximal length of its edges (times some constant).
The approach follows the one from [19], mass transportation and convexity properties of the relative entropy being studied along h-geodesics. The work [6] continues this study towards a rough curvature-dimension condition, which brings more geometrical consequences.
We present here a survey of these results concerning generalized Ricci curvature bounds for metric measure spaces. In Section 2 we follow the works
of K.T. Sturm [19] and [20] where the lower Ricci curvature bounds and the curvature-dimension condition have been introduced and studied. The mate- rial is too large to be covered here, we just point out the main definitions and properties.
In Section 3 we present the lower rough curvature bounds for metric measure spaces, with emphasize on discrete spaces and graphs, following [5].
The main two results (Theorem 3.6) and Theorem 3.7) state the stability of the rough curvature bounds under D-convergence, respectively under taking discretizations. The latter one provides a series of concrete examples, among them the homogeneous planar graphs. It is also related to some notions of combinatorial curvature, see e.g. [9], [12], [11], [7]. Positive rough curvature bound implies a perturbed transportation cost inequality, weaker than what is usually called the Talagrand inequality. However, it still implies concentra- tion of the reference measure m and exponential integrability of the Lipschitz functions with respect to m.
The fourth section introduces the rough curvature-dimension conditionh- CD(K, N) for metric measure spaces, coming with an additional upper bound for the “dimension”, following [6]. The planar graphs for instance are dis- crete analogues of connected Riemannian surfaces, therefore they deserve to be considered 2-dimensional discrete spaces. Besides, an upper bound for the dimension would be expected to bring, by analogy with the finite dimen- sional Riemannian manifolds, more geometrical consequences in our discrete setting. The rough curvature bound can be regarded as a limit case or as anh- CD(K,∞) rough curvature-dimension condition. The stabilities issues under D-convergence and discretizations is also mentioned.
2. CURVATURE BOUNDS
FOR GEODESIC METRIC MEASURE SPACES
In the following, we will call metric measure space a triple (M,d, m), where (M,d) is a complete and separable metric space and m is a measure on M, equipped with its Borelσ-algebra B(M); m is supposed to be locally finite in the sense that m(Br(x))<∞ for all x∈M and all sufficiently small r >0. We say that the metric measure space (M,d, m) is normalized ifm is a probability measure (m(M) = 1).
According to [19], two metric measure spaces (M,d, m) and (M0,d0, m0) are called isomorphic iff there exists an isometryψ : M0 → M00 between the supports M0 := supp[m] ⊂ M and M00 := supp[m0] ⊂ M0, with ψ∗m = m0. The diameterof a metric measure space (M,d, m) will be the diameter of the metric space (supp[m], d).
We shall use the notion of L2-transportation distance D for two metric measure spaces (M,d, m) and (M0,d0, m0), as defined in [19]:
D((M,d, m),(M0,d0, m0)) = inf Z
MtM0
ˆd2(x, y)dq(x, y) 1/2
,
where ˆd ranges over all couplings of d and d0 and q ranges over all couplings of m and m0. We recall that a measure q on the product space M ×M0 is a coupling of m and m0 if q(A×M0) =m(A) and q(M ×A0) =m0(A0) for all measurable A⊂M, A0 ⊂M0; a pseudo-metric ˆdon the disjoint unionMtM0 is a coupling of d and d0 iff ˆd(x, y) = d(x, y) and ˆd(x0, y0) = d0(x0, y0) for all x, y ∈ supp[m]⊂ M and all x0, y0 ∈ supp[m0] ⊂ M0. The L2-transportation distance Ddefines a complete and separable length metric on the family of all isomorphism classes of normalized metric measure spaces (M,d, m) that satisfy the condition R
Md2(y, x)dm(x) < ∞ for some (hence all) y ∈ M. The D- convergence is closely related to the measured Gromov-Hausdorff convergence introduced in [8].
A sequence of compact and normalized metric measure spaces denoted {(Mn,dn, mn)}n∈Nconverges in the sense ofmeasured Gromov-Hausdorff con- vergence (briefly, mGH-converges) to a compact normalized metric measure space (M,d, m) iff there exist a sequence of numbers n & 0 and a se- quence of measurable maps fn : Mn → M such that for all x, y ∈ Mn,
|d(fn(x), fn(y))−dn(x, y)| ≤ n, for any x ∈ M there exists y ∈ Mn with d(fn(y), x) ≤n and such that (fn)∗mn →m weakly on M for n→ ∞. Ac- cording to Lemma 3.17 in [19], any mGH-convergent sequence of normalized metric measure spaces is also D-convergent; furthermore, for any sequence of normalized compact metric measure spaces having full supports and uni- form bounds for the doubling constants and for the diameters, the notion of mGH-convergence is equivalent to the one of D-convergence.
One can easily notice thatD((M,d, m),(M0,d0, m0)) = inf ˆdW(φ∗m,φ0∗m0) where the inf is taken over all metric spaces ( ˆM ,d) with isometric embeddingsˆ φ : M0 ,→ Mˆ, φ0 : M00 ,→ Mˆ of the supports M0 and M00 of m and m0, respectively, and where ˆdW denotes theL2-Wasserstein distance derived from the metric ˆd.
Recall that for any metric space (M,d) theL2-Wasserstein distance be- tween two measures µand ν onM is defined as
dW(µ, ν) = inf (Z
M×M
d2(x, y)dq(x, y) 1/2
:q is a coupling ofµand ν )
, with the convention inf∅=∞. For further details about the Wasserstein dis- tance see the excellent monographs [22] and [23]. We shall denote byP2(M,d)
the space of all probability measures ν which have finite second moments R
Md2(y, x)dν(x)<∞ for some (hence all) y∈M.
For a given metric measure space (M,d, m) we setP2(M,d, m) the space of all probability measuresν ∈ P2(M,d) which are absolutely continuous with respect to m. If ν =ρ·m ∈ P2(M,d, m) we consider the relative entropy of ν with respect to m defined by Ent(ν|m) := lim
&0
R
{ρ>}ρlogρdm. We denote by P2∗(M,d, m) the subspace of measures ν ∈ P2(M,d, m) of finite entropy Ent(ν|m)<∞.
In classical Riemannian geometry, given a pointx in a Riemannian ma- nifold, the Ricci curvature Ricx is defined on the tangent spaceTxM as
Ricx(v, v) := trace{w→ R(v, w)v}, v∈TxM,
where Ris the curvature tensor. The Ricci curvature Ricx measures the non- euclidian behavior of the manifold in direction v.
In the paper [18] the authors give the following characterization of the Ricci curvature bound for Riemannian manifolds.
Theorem2.1. For any smooth connected Riemannian manifoldM with intrinsic metricdand volume measurem and anyK∈Rthe following proper- ties are equivalent:
(i) Ricx(v, v)≥K|v|2 for x∈M and v∈Tx(M).
(ii) The entropy Ent(·|m) is displacement K-convex on P2(M) in the sense that for each geodesic γ : [0,1]→ P2(M) and for each t∈[0,1]
Ent(γ(t)|m)≤(1−t) Ent(γ(0)|m) +tEnt(γ(1)|m)− K2t(1−t)d2W(γ(0), γ(1)).
This characterization makes no use of the differentiability issue and con- dition (ii) can be posed in any (geodesic) metric measure space. Therefore, (ii) might stand as a definition for a lower Ricci curvature bound. Indeed [19] proves stability under D-convergence and pinpoints a series of results that correspond to classical theorems from Riemannian geometry involving Ricci curvature.
We recall here the definitions of the lower curvature bounds for metric measure spaces introduced in [19]:
Definition 2.2. (i) A metric measure space (M,d, m) has curvature≥K for some number K∈R iff the relative entropy Ent(·|m) is weakly K-convex onP2∗(M,d, m) in the sense that for each pairν0, ν1 ∈ P2∗(M,d, m) there exists a geodesic Γ : [0,1]→ P2∗(M,d, m) connectingν0 and ν1 with
Ent(Γ(t)|m)≤(1−t) Ent(Γ(0)|m) +tEnt(Γ(1)|m)−
(2.1)
−K
2 t(1−t)d2W(Γ(0),Γ(1)) for all t∈[0,1].
(ii) The metric measure space (M,d, m) has curvature ≥ K in the lax sense iff for each >0 and for each pair ν0, ν1 ∈ P2∗(M,d, m) there exists an -midpoint η∈ P2∗(M,d, m) ofν0 andν1 with
(2.2) Ent(η|m)≤ 1
2Ent(ν0|m) +1
2Ent(ν1|m)−K
8d2W(ν0, ν1) +. We write brieflyCurv(M,d, m)≥K, resp. Curvlax(M,d, m)≥K.
A pointy in a given metric space (M,d) is an -midpointof x0 and x1 if d(xi, y)≤ 12d(x0, x1) +for each i= 0,1. We call y midpoint of x0 and x1 if d(xi, y)≤ 12d(x0, x1) for i= 0,1.
IfM is compact then the curvature bounds in the usual sense and in the lax sense coincide, according to Lemma 4.7 in [19].
In Riemannian setting, the following result holds:
Theorem 2.3. Let M be a complete Riemannian manifold having Rie- mannian distance d and Riemannian volume m, and put m0 = exp(−V)·m for a C2 function V :M →R. Then
Curv(M,d, m0) = inf{RicM(ξ, ξ) + HessV(ξ, ξ) :ξ2∈T M,|ξ|= 1}. In particular, (M,d, m) has curvature ≥K if and only if the Ricci curvature of M is ≥K.
In the next proposition we collect several properties of the curvature bounds from [19], which show their behavior under various transformations (isomorphisms, scaling, weights, subsets, products).
Proposition 2.4. (i) If (M,d, m) and (M0,d0, m0) are two isomorphic metric measure spaces, then
Curv(M,d, m) =Curv(M0,d0, m0).
For any metric measure space (M,d, m) and α, β >0 Curv(M, αd, βm) =α−2Curv(M,d, m).
(ii)For any metric measure space (M,d, m) and any lower bounded and measurable function V :M →R
Curv(M,d, e−V)≥Curv(M,d, m) + HessV, where HessV := sup{K ∈R:V isK-convex on supp[m]}.
(iii) If (M,d, m) is a metric measure space and M0 is a convex subset of M, then
Curv(M0,d, m)≥Curv(M,d, m).
(iv)Let (Mi,di, mi) for i= 1, . . . , n be metric measure spaces and (M,d, m) =
n
O
i=1
(Mi,di, mi).
Assume that M is nonbranching and compact. Then Curv(M,d, m) = inf
i∈{1,...,n}Curv(Mi,di, mi).
Similar properties hold for Curvlax instead ofCurv.
The notion of lower curvature bound described above is dimension in- dependent. In order to obtain further results one has to analyze a curvature- dimension condition CD(K, N) involving two parameters: K playing the role of a generalized lower bound for the Ricci curvature, and N playing the role of an upper bound for the dimension. In this way, the curvature bound Curv(M,d, m)≥K can be understood as aCD(K,∞) condition. The curva- ture-dimension condition for geodesic metric measure spaces has been intro- duced and analyzed in the paper [20].
In the same framework of a metric measure space (M,d, m), a point z in M is called a t-intermediate point between x and y for some t ∈ [0,1] if d(x, z) =t·d(x, y) and d(z, y) = (1−t)·d(x, y).
Instead of the relative entropy Ent(·|m) one should use the R´enyi entropy functional, which depends also on a parameter N ≥ 1 that plays the role of the dimension in the following material. The R´enyi entropy functional with respect to our reference measure m is defined as
SN(·|m) :P2(M,d)→R with
SN(ν|m) :=− Z
M
ρ−1/Ndν,
where ρ is the density of the absolutely continuous part νc with respect to m in the Lebesgue decomposition ν = νc +νs = ρm+νs of the measure ν ∈ P2(M,d).
Lemma 1.1 from [20] states that
Lemma 2.5. Assume that m(M) is finite.
(i) For each N >1 the R´enyi entropy functional SN(·|m) is lower semi- continuous and satisfies
−m(M)1/N ≤SN(·|m)≤0 onP2(M,d).
(ii)For any ν ∈ P2(M,d) Ent(·|m) = lim
N→∞N(1 +SN(ν|m)).
For given K, N ∈ R with N ≥ 1 and (t, θ) ∈ [0,1]×R+ we use the notation
τK,N(t) (θ) =
∞ ifKθ2 ≥(N−1)π2, tN1 sin
q K N−1tθ
sin q K
N−1θ
!1−N1
if 0< Kθ2 <(N−1)π2,
t ifKθ2 = 0 or
ifKθ2 <0 and N = 1, tN1 sinh
q−K N−1tθ
sinh q−K
N−1θ
!1−N1
ifKθ2 <0 and N >1.
Remark 2.6. For arbitrarily fixed t∈(0,1) and θ∈ (0,∞) the function (K, N)→τK,N(t) (θ) is continuous, nondecreasing inKand non-increasing inN. The curvature-dimension condition for geodesic spaces (M,d, m) has been introduced in [20] in the following way:
Definition 2.7. Given two numbers K, N ∈ R with N ≥1 we say that a metric measure space (M,d, m) satisfies the curvature-dimension condition CD(K, N) iff for each pairν0,ν1 ∈ P2(M,d, m) there exist an optimal coupling q ofν0,ν1 and a geodesic Γ : [0,1]→ P2(M,d, m) connecting ν0,ν1 and with
SN0(ηt|m)≤ − Z h
τK,N(1−t)0(d(x0, x1))·ρ−1/N
0
0 (x0) + (2.3)
+ τK,N(t) 0(d(x0, x1))·ρ−1/N1 0(x1) i
dq(x0, x1)
for all t∈[0,1] and all N0 ≥N. Hereρi denotes the density functions of the absolutely continuous parts of νi with respect to m,i= 1,2.
If (M,d, m) has finite mass and satisfies the curvature-dimension condi- tion CD(K, N) for some K and N then it has curvature ≥K in the sense of Definition 2.2. In other words, the conditionCurv(M,d, m)≥K may be inter- preted as the curvature-dimension conditionCD(K,∞) for the space (M,d, m).
For Riemannian manifolds the curvature-dimension conditionCD(K, N) reverts to “Ricci curvature bounded below byKand dimension bounded above by N”, as it is shown in Theorem 1.7 from [20]:
Theorem 2.8. Let M be a complete Riemannian manifold with Rie- mannian distance d and Riemannian volume m and let numbers K, N ∈ R with N ≥1 be given.
(i) The space (M,d, m) satisfies the condition CD(K, N) if and only if the Riemannian manifold has Ricci curvature ≥K and dimension≤N.
(ii) Moreover, in this case for every measurable function V : M → R the weighted space (M,d, V ·m) satisfies the curvature-dimension condition
CD(K+K0, N +N0) provided
HessV1/N0 ≤ −K0
N0 ·V1/N0 for some numbers K0∈R,N0 >0 in the sense that
V(γt)1/N0 ≥σ(1−t)K0,N0(d(γ0, γ1))V(γ0)1/N0+σK(t)0,N0(d(γ0, γ1))V(γ1)1/N0 for each geodesic γ : [0,1]→M and eacht∈[0,1]. Here we denoted
σK,N(t) (θ) := sin rK
Ntθ
!, sin
rK Nθ
!
if 0< Kθ2< N π2, and with appropriate modifications otherwise.
The curvature-dimension conditions has various geometric consequences, like Brunn-Minkowski inequality, Bishop-Gromov volume growth estimate, and Bonnet-Myers theorem.
3. ROUGH CURVATURE BOUNDS FOR METRIC MEASURE SPACES
The approach presented in the previous chapter required the Wasserstein space of probability measures (and thus in turn the underlying space) to be a geodesic space. Therefore, in the original form it cannot apply to discrete spaces. Besides, metric graphs would have no lower curvature bound in the sense of [19], since the vertices are branch points of geodesics, which can- not allow the K-convexity of the entropy. Therefore, mass transportation and convexity properties of the relative entropy must be studied alongh-geodesics instead of geodesics. In this respect, we present in this chapter some results from [5].
We refer in the sequel to the larger class ofh-geodesic spaces, as natural generalizations of geodesic spaces:
Definition 3.1. For a given h >0, we say that a metric space (M,d) is h-geodesic iff for any pair of points x0, x1 ∈M and any t∈[0,1] there exists xt∈M with
(3.1) d(x0, xt)≤td(x0, x1) +h, d(xt, x1)≤(1−t)d(x0, x1) +h.
The point xt will be referred to as the h-rough t-intermediate point between x0andx1. Theh-rough 1/2-intermediate point is theh-midpoint ofx0 andx1. There are many examples ofh-geodesic spaces that are not geodesic. For instance, any nonempty set X, equipped with the discrete metricd(x, y) = 0 for x=y and 1 for x6=y, is h-rough geodesic for each h≥1/2. In this case,
any point is an h-midpoint of any pair of distinct points. Another example:
the space (Rn,d) with the metric d(x, y) = |x−y| ∧, for a fixed > 0, is h-rough geodesic forh≥/2 (here | · |is the euclidian metric).
The spaces that we focus on are actually the various discrete spaces and also some geodesic spaces with branch points, e.g. graphs, that do not have curvature bounds as defined in [19].
For a discrete h-rough geodesic metric space (M,d) the parameter h should be understood as a discretization size or “resolution” of the metric space M. In anh-geodesic space an arbitrary pair of pointsxandyis not necessarily connected by a geodesic but by a chain of points x=x0, x1, . . . , xn=y which have the intermediate distances less then h/2.
We make use of two types of perturbations of the Wasserstein distance, defined as follows:
Definition 3.2. Let (M,d) be a metric space. For each h > 0 and each pair of measures ν0,ν1 ∈ P2(M,d) define
(3.2) d±hW (ν0, ν1) := inf (Z
[(d(x0, x1)∓h)+]2dq(x0, x1) 1/2)
, where q ranges over all couplings of ν0 and ν1 and (·)+ denotes the posi- tive part.
According to Theorem 4.1 from [23] there exists a coupling for which the infimum in (3.2) is attaint. We will call it +h-optimal coupling (resp.
−h-optimal coupling) of ν0 and ν1.
The two perturbationsd+hW andd−hW satisfy the following properties:
Lemma 3.3. For any h >0 we have:
(i) d+hW ≤dW ≤d+hW +h;
(ii)dW ≤d−hW ≤dW +h.
Moreover, if 0 < h1 < h2 are arbitrarily given, then for each pair of probabilities ν0 andν1
(iii) d−hW 1(ν0, ν1)<d−hW2(ν0, ν1);
(iv) d+hW 1(ν0, ν1)≥d+hW2(ν0, ν1) with strict inequality if and only if d+hW1 (ν0, ν1)>0.
We introduce now the notion of lower rough curvature bound:
Definition 3.4. A metric measure space (M,d, m) has h-rough curva- ture ≥ K for some numbers h > 0 and K ∈ R iff for each pair ν0, ν1 ∈ P2∗(M,d, m) and for anyt∈[0,1] there exists anh-rought-intermediate point ηt∈ P2∗(M,d, m) betweenν0 and ν1 satisfying
(3.3) Ent(ηt|m)≤(1−t) Ent(ν0|m) +tEnt(ν1|m)−K
2t(1−t)d±hW (ν0, ν1)2,
where the sign ind±hW (ν0, ν1) is chosen ‘+’ ifK >0 and ‘−’ if K <0. Briefly, we write in this case h-Curv(M,d, m)≥K.
Remark 3.5. (i) If (M,d, m), (M0,d0, m0) are isomorphic metric measure spaces andK∈Rthenh-Curv(M,d, m)≥K if and only ifh-Curv(M0,d0, m0)
≥K.
(ii) If (M,d, m) is a metric measure space and α, β > 0 then we have h-Curv(M,d, m) ≥ K if and only if αh-Curv(M, αd, βm) ≥ αK2, because Ent(ν|βm) = Ent(ν|m)−logβ, (α ·d)±hW (ν0, ν1) = α ·d±hW (ν0, ν1) and for t ∈[0,1] ηt is h-rough t-intermediate point between µ, ν with respect to dW if and only if ηt is αh-rough t-intermediate point between µ, ν with respect to (αd)W.
The lower rough curvature bounds have the following stability property:
Theorem3.6. We consider(M,d, m)a normalized metric measure space and assume that {(Mh,dh, mh)}h>0 is a family of normalized metric mea- sure spaces with uniformly bounded diameter and with the curvature bound h-Curv(Mh,dh, mh)≥Kh for Kh→K as h→0. If
(Mh,dh, mh)−→D (M,d, m) as h→0 then
Curvlax(M,d, m)≥K.
If in addition M is compact then
Curv(M,d, m)≥K.
Proof. Let {(Mh,dh, mh)}h>0 be a family of normalized discrete metric measure spaces. Assume that we have suph>0diam(Mh,dh, mh), diam(M,d, m) ≤ ∆ and (Mh,dh, mh) −→D (M,d, m) as h → 0. Let now > 0 and ν0=ρ0m,ν1 =ρ1m∈ P2∗(M,d, m) be given. ChooseR with
(3.4) sup
i=0,1
Ent(νi|m) +|K|
8 ∆2+
8[∆2+ 3|K|(2∆ + 3)]≤R.
We want to deduce the existence of an-midpointη which satisfies inequality (2.2). Therefore, choose 0< h < with|Kh−K|< and
(3.5) D((Mh,dh, mh),(M,d, m))≤exp
−2 + 4∆2R 2
.
We define the canonical maps Q0h:P2(M,d, m)→ P2(Mh,dh, mh), Qh : P2(Mh,dh, mh)→ P2(M,d, m) like in Subsection 4.5 in [19]:
Consider qh a coupling of m and mh and ˆdh a coupling of d and dh such that
Z
ˆd2h(x, y) dqh(x, y)≤2D2((M,d, m),(Mh,dh, mh)).
Let Q0h and Qh be the disintegrations of qh w.r.t. mh and m, resp., that is dqh(x, y) = Q0h(y,dx)dmh(y) = Qh(x,dy)dm(x) and let ˆ∆ denote the m- essential supremum of the map
x7→
Z
Mh
ˆd2h(x, y)Qh(x,dy) 1/2
. In our case ˆ∆≤2∆.
For ν =ρm ∈ P2(M,d, m) define Q0h(ν)∈ P2(Mh,dh, mh) by Q0h(ν) :=
ρhmh where
ρh(y) :=
Z
M
ρ(x)Q0h(y,dx).
The map Qh is defined similarly. Lemma 4.19 from [19] gives the following estimates
(3.6) Ent(Q0h(ν)|mh)≤Ent(ν|m) for allν =ρ m and
(3.7) d2W(ν, Q0h(ν))≤ 2 + ˆ∆2·Ent(ν|m)
−logD((M,d, m),(Mh,dh, mh)).
provided D((M,d, m),(Mh,dh, mh))<1. Analogous estimates hold forQh. For our givenν0 =ρ0m,ν1 =ρ1m∈ P2∗(M,d, m) put
νi,h:=Q0h(νi) =ρi,hmh withρi,h(y) =R
ρi(x)Q0h(y,dx) fori= 0,1 and letηh be anh-midpoint ofν0,h and ν1,h such that
(3.8) Ent(ηh|mh)≤ 1
2Ent(ν0,h|mh) +1
2Ent(ν1,h|mh)−Kh
8 dδWhh(ν0,h, ν1,h)2, where δh is the sign ofKh.
From (3.5)–(3.7), we conclude
d2W(ν0, ν0,h)≤ 2 + ˆ∆2·Ent(ν0|m)
−logD((M,d, m),(Mh,dh, mh))
≤ 2 + 4∆2R
−logD((M,d, m),(Mh,dh, mh)) ≤2 and similarly d2W(ν1, ν1,h)≤2.
IfK <0 we can supposeKh <0, too. From Lemma 3.3 (ii) we have d−hW (ν0,h, ν1,h)2≤(dW(ν0,h, ν1,h) +h)2≤(dW(ν0, ν1) + 3)2
≤dW(ν0, ν1)2+ 6∆ + 92, because dW(ν0, ν1)≤∆.
For K > 0 one can choose h small enough to ensure Kh > 0. Then Lemma 3.3 (i) implies
dW(ν0, ν1)2 ≤(dW(ν0,h, ν1,h) + 2)2≤
d+hW (ν0,h, ν1,h) + 32
≤d+hW (ν0, ν1)2+ 6∆ + 92.
In both cases the estimates above combined with (3.6), (3.8) and the fact that we chose h with−Kh < −K will imply
(3.9) Ent(ηh|mh)≤ 1
2Ent(ν0|m) +1
2Ent(ν1|m)−K
8d2W(ν0, ν1) +0 with 0 =[∆2+ 3|K|(2∆ + 3)]/8.
The caseK= 0 follows by the calculations above, depending on the sign of Kh.
Finally, put
η =Qh(ηh).
Then again by (3.5), the estimates given in Lemma 4.19 of [19] forQh and by the previous estimate (3.9) for Ent(ηh|mh) we deduce
d2W(ηh, η)≤ 2 + ˆ∆2·Ent(ηh|mh)
−logD((M,d, m),(Mh,dh, mh))
≤ 2 + 4∆2R
−logD((M,d, m),(Mh,dh, mh)) ≤2.
Fori= 0,1 we havedW(η, νi)≤2+dW(ηh, νi,h)≤2+12dW(ν0,h, ν1,h) +h≤
1
2dW(ν0, ν1) + 4. Hence, sup
i=0,1
dW(η, νi)≤ 1
2dW(ν0, ν1) + 4, i.e., η is a (4)-midpoint ofν0 and ν1. Furthermore, by (3.6),
Ent(η|m)≤Ent(ηh|mh)≤ 1
2Ent(ν0|m) +1
2Ent(ν1|m)−K
8 d2W(ν0, ν1) +0 with 0 as above, which proves that Curvlax(M,d, m)≥K.
The lower curvature bounds are also stable under taking discretizations.
For a given metric measure space (M,d, m) we define its discretizations as follows: forh >0 considerMh a discrete subset ofM, sayMh ={xn:n∈N},
withM =
∞
S
i=1
BR(xi), whereR=R(h)&0 ash&0. We chooseAi ⊂BR(xi) mutually disjoint subsets withxi ∈Ai,i∈Nand
∞
S
i=1
Ai =M, and consider the measure mh on Mh given by mh({xi}) :=m(Ai),i∈N. We call (Mh,d, mh) a discretizationof (M,d, m). We have then the following stability result:
Theorem 3.7. (i) If m(M) < ∞ then (Mh,d, mh) −→D (M,d, m) as h→0.
(ii)If Curvlax(M,d, m)≥K withK6= 0then for each h >0 and for each discretization (Mh,d,mh) withR(h)< h/4 we have h-Curv(Mh,d,mh)≥K.
(iii) If Curv(M,d, m)≥K for some real number K then for eachh >0 and for each discretization (Mh,d, mh) with R(h) ≤ h/4 we have the bound h-Curv(Mh,d, mh)≥K.
Proof. (i) The measure q =
∞
P
i=1
(m(Ai)δxi)×(1Aim) is a coupling of mh and m, so
D2((Mh,d, mh),(M,d, m))≤ Z
Mh×M
d2(x, y) dq(x, y)
=
∞
X
i=1
m(Ai) Z
Ai
d2(xi, y) dm(y)
≤
∞
X
i=1
m(Ai)2
!
R(h)2≤R(h)2
∞
X
i=1
m(Ai)
!2
=R(h)2m(M)2 →0 ash→0.
(ii) We fix h >0 and consider a discretization (Mh,d, mh) of (M,d, m) withR(h)< h/4. Letν0h, ν1h ∈ P2∗(Mh,d, mh) be given; it is enough to give the proof forν0h, ν1h with compact support. Suppose thenνih= n
P
j=1
αhi,j1{xj} mh, i = 1,2 (some of the αhi,j could be zero). Let us take also an arbitrary t ∈ [0,1]. Define νi := n
P
j=1
αhi,j1Aj
m ∈ P2∗(M,d, m) for i = 1,2. Choose >0 such that
(3.10) 4R(h) +≤h.
Since Curvlax(M,d, m)≥K for our given t∈[0,1] there is ηt∈ P2∗(M,d, m) an -rough t-intermediate point betweenν0 andν1 such that
(3.11) Ent(ηt|m)≤(1−t) Ent(ν0|m) +tEnt(ν1|m)−K
2t(1−t)d2W(ν0, ν1) +.
We can compute Ent(νi|m) =
n
X
j=1
Z
Aj
αhi,jlogαhi,jdm=
n
X
j=1
αhi,jlogαhi,jmh({xj}) (3.12)
= Ent(νih|mh),
fori= 0,1. Denoteηht({xj}) :=ηt(Aj),j= 1,2, . . . , nand supposeηt=ρt·m.
From Jensen’s inequality we get Ent(ηth|mh) =
∞
X
j=1
R
Ajρtdm m(Aj) log
R
Ajρtdm
m(Aj) mh({xj})
≤
∞
X
j=1
1 m(Aj)
Z
Aj
ρtlogρtdm
!
mh({xj}) = Ent(ηt|m), which together with (3.11) and (3.12) yields
(3.13) Ent(ηth|mh)≤(1−t) Ent(ν0h|mh)+tEnt(ν1h|mh)−K
2t(1−t)d2W(ν0, ν1)+.
Let us consider now the caseK <0. Ifqh is a−2R(h)-optimal coupling of ν0h and ν1h, then the formula
bq:=
n
X
j,k=1
qh({(xj, xk)})δ(xj,x
k)× 1Aj×Ak
m(Aj)m(Ak)(m×m)
defines a measure on Mh×Mh×M×M which has marginalsν0h,ν1h,ν0 and ν1. Furthermore, the projection of qbon the first two factors is equal to qh. Therefore, we have
dW(ν0, ν1)2 ≤ Z
d(x, y)2dq(xb h, yh, x, y)
≤ Z h
d(x, xh) +d(xh, yh) +d(yh, y)i2
dq(xb h, yh, x, y)
=
n
X
j,k=1
qh({(xj, xk)}) m(Aj)m(Ak)
Z
Aj×Ak
[d(x, xj) +d(xj, xk) +d(xk, y)]2dm(x)dm(y)
≤
n
X
j,k=1
qh({(xj, xk)}) (d(xj, xk) + 2R(h))2=d−2R(h)W (ν0h, ν1h)2,
which together with (3.13) gives
Ent(ηht|mh)≤(1−t) Ent(ν0h|mh) +tEnt(ν1h|mh)−
(3.14)
−K
2 t(1−t)d−2R(h)W (ν0h, ν1h)2+.
In the case K > 0 we consider an optimal coupling q of ν0 and ν1 and we show that the measure
qeh :=
n
X
j,k=1
q(Aj×Ak)δ(xj,xk)
is a coupling of ν0h and ν1h. Indeed, if A⊂Mh then we have in turn
n
X
j,k=1
q(Aj ×Ak)δ(xj,xk)(A×Mh) =
n
X
j,k=1
q(Aj×Ak)δxj(A)
=
n
X
j=1
q(Aj×M)δxj(A) =
n
X
j=1
ν0(Aj)δxj(A) =
n
X
j=1
ν0h({xj})δxj(A) =ν0h(A).
Since for any j, k = 1,2, . . . , n and for arbitrary x ∈Aj and y ∈ Ak we have (d(xj, xk)−2R(h))+ ≤ (d(xj, xk)−d(x, xj)−d(y, xk))+ ≤ d(x, y) one can estimate
d+2R(h)W (ν0h, ν1h)2 ≤
n
X
j,k=1
q(Aj×Ak) [(d(xj, xk)−2R(h))+]2
=
n
X
j,k=1
Z
Aj×Ak
[(d(xj, xk)−2R(h))+]2dq(x, y)
≤
n
X
j,k=1
Z
Aj×Ak
[(d(xj, xk)−d(x, xj)−d(y, xk))+]2dq(x, y)
≤
n
X
j,k=1
Z
Aj×Ak
d(x, y)2dq(x, y) = Z
M×M
d(x, y)2dq(x, y)
=dW(ν0, ν1)2. Therefore, from (3.13) we get
Ent(ηht|mh)≤(1−t) Ent(ν0h|mh) +tEnt(ν1h|mh)−
(3.15)
−K
2 t(1−t)d+2R(h)W (ν0h, ν1h)2+. Forsmall enough, we can obtain
(3.16) −K
2t(1−t)d±2R(h)W (ν0h, ν1h)2+≤ −K
2t(1−t)d±hW (ν0h, ν1h)2
and then (3.14), (3.15) yield
Ent(ηht|mh)≤(1−t) Ent(ν0h|mh) +tEnt(ν1h|mh)−
(3.17)
−K
2t(1−t)d±hW (ν0h, ν1h)2,
depending on the sign ofK. The inequality (3.16) fails only when K >0 and d+hW (ν0h, ν1h) = 0, but in this casedW(ν0h, ν1h)≤h and eitherη =ν0h orη =ν1h verifies directly the condition (3.3) from the definition of h-rough curvature bound for the discretization.
Obviously, the measureπ=
n
P
j=1
ηht({xj})δxj×1Ajηt
is a coupling ofηht and ηt, therefore,
d2W(ηth, ηt)≤ Z
Mh×M
d2(x, y) dπ(x, y)≤R2(h),
and similarly d2W(νih, νi) ≤ R2(h) for i = 1,2. Since the measure ηt is an -rough t-intermediate point betweenν0 and ν1 we deduce
dW(ηth, ν0h)≤dW(ηt, ν0) + 2R(h)≤tdW(ν0, ν1) + 2R(h) +
≤tdW(ν0h, ν1h) + 2R(h)(1 +t) + and by a similar argument
dW(ηth, ν1h)≤(1−t)dW(ν0h, ν1h) + 2R(h)(2−t) +.
Using (3.10), we conclude thatηh is anh-rough t-intermediate point between ν0h and ν1h, which together with (3.17) proves thath-Curv(Mh,d, mh)≥K.
(iii) The proof is similar to (ii).
The above result gives a large class of examples:
Example1. The spaceZn equipped with the metric d1 coming from the norm|·|1inRn, defined by|x|1 =
n
P
i=1
|xi|, and with the measuremn= P
x∈Zn
δx, has h-Curv(Zn,d1, mn)≥0 for any h≥2n.
Example 2. The n-dimensional grid En having Zn as set of vertices, equipped with the graph distance and with the measure mn, which is the 1-dimensional Lebesgue measure on the edges, hash-Curv(En,d1, mn)≥0 for any h≥2(n+ 1).
Example3. The graph G that tiles the euclidian plane with equilateral triangles of edger, endowed with the graph metricdGinduced by the euclidian metric and with the 1-dimensional Lebesgue measure m on the edges, hash- curvature ≥0 for any h≥8r√
3/3.
Example4. LetG0 be the graph that tiles the euclidian plane with regular hexagons of edge length r, equipped also with the graph metric dG0 and with the 1-dimensional measurem0. ThenG0 hash-curvature≥0 for anyh≥34r/3.
Example5 (Planar graphs). Consider the (possibly infinite) homogeneous graph G(l, n, r) with vertices of constant degree l≥3, with faces bounded by polygons withn≥3 edges and such that all edges have the same lengthr >0.
We have then the following embedding result:
Lemma 3.8. (i) If 1l + n1 < 12 then G(l, n, r) can be embedded into the 2-dimensional hyperbolic space with constant sectional curvature
(3.18) K =−1
r2
"
arccosh 2cos2 πn sin2 πl −1
!#2
.
There are infinitely many choices of such l and n. In any case, the graph is unbounded.
(ii) If 1l + n1 > 12 then G(l, n, r) is one of the five regular polyhedra (Tetrahedron, Octahedron, Cube, Icosahedron, Dodecahedron) and can be em- bedded into the 2-dimensional sphere with constant sectional curvature
(3.19) K= 1
r2
"
arccos 2cos2 πn sin2 πl −1
!#2
.
(iii)If 1l+n1 = 12 thenG(l, n, r)can be embedded into the euclidian plane (K = 0). In this case there are exactly three cases corresponding to the 3 regular tessellations of the euclidian plane: the tessellation of triangles (l= 6, n= 3), of squares(l=n= 4), and of hexagons(l= 3, n= 6).
Then the dual graph G(l, n, r)∗ = G(n, l, r0) is embedded into the 2- manifold of the same constant curvature as G(l, n, r), where the dual edge length is
r0 :=r·arccosh 2cos2 πn sin2 πl −1
! ,
arccosh 2cos2 πl sin2 πn −1
!
forK <0 and with appropriate modifications for the other two cases.
In each of the three cases from Lemma 3.8 the 2-manifold can be equipped with the intrinsic metric d and with the Riemannian volume vol. We endow G(l, n, r) with the metricd induced by the corresponding Riemannian metric and with the uniform measuremon the edges. We denote further byV(l, n, r) the set of vertices of the graph G(l, n, r) equipped with the same metric d inherited from the Riemannian manifold and with the counting measure
me :=X
v∈V
δv.
Proposition 3.9. For any numbers l, n ≥ 3 and for any r > 0 both metric measure spaces (V(l, n, r),d,m)e and (G(l, n, r),d, m) have h-curvature
≥K for h≥r·C(l, n), where
(3.20) K =
−r12
arccosh
2cos
2(πn)
sin2(πl) −1 2
for 1l +n1 > 12,
1 r2
arccos
2cos
2(πn)
sin2(πl) −1 2
for 1l +n1 < 12,
0 for 1l +n1 = 12,
and
C(l, n) = 4·arcsinh 1 sin πn
scos2 πn sin2 πl −1
!,
arccosh 2cos2 πn sin2 πl −1
! . In the literature there are various notions of combinatorial curvature for graphs, see for instance [9], [11], [7]. The combinatorial curvature intro- duced by Gromov in [9] was used in studying hyperbolic groups. Later on it was modified and investigated by Higuchi [11] and other authors. Forman has introduced in [7] a different notion of combinatorial Ricci curvature for cell complexes. The graphs considered in the above mentioned works have neither specified metric, nor specified reference measure.
If (M, d) is a metric space and m ∈ P2(M,d) is a given probability measure, mis said to satisfy a Talagrand inequality (or a transportation cost inequality) with constant K iff for allν ∈ P2(M,d)
(3.21) dW(ν, m)≤
r2 Ent(ν|m)
K .
Such an inequality was first proved by Talagrand in [21] for the canonical Gaussian measure on Rn. Under a positive rough curvature bound a weaker inequality holds, in terms of the perturbationd+hW of the Wasserstein distance:
Proposition3.10 (“h-Talagrand Inequality”). Let(M,d, m)be a metric measure space which has h-Curv(M,d, m) ≥K for h > 0 and K > 0. Then for each ν ∈ P2(M,d) we have
(3.22) d+hW (ν, m)≤
r2 Ent(ν|m)
K .
We will call (3.22) h-Talagrand inequality.
Proof. See Proposition 6.1 from [5].
A Talagrand inequality for the measuremimplies a concentration of mea- sure inequality for m(see for instance [15]). A weakerh-Talagrand Inequality is still good enough to provide concentration of measure.
For a given Borel setA⊂M denote the (open)r-neighborhood of A by Br(A) := {x ∈ M : d(x, A) < r} for r > 0. The concentration function of (M,d, m) is defined as
α(M,d,m)(r) := sup
1−m(Br(A)) :A∈ B(M), m(A)≥ 1 2
, r >0.
We refer to [13] for further details on measure concentration.
The following result shows that positive rough curvature bound implies a normal concentration inequality, via h-Talagrand inequality.
Proposition 3.11. Let (M,d, m) be a metric measure space such that h-Curv(M,d, m) ≥K >0 for some h > 0. Then there exists an r0 >0 such that for all r ≥r0
α(M,d,m)(r)≤e−Kr2/8. Proof. See Proposition 6.2 from [5].
According to [4], a Talagrand type inequality implies exponential inte- grability of the Lipshitz functions. One can prove further that anh-Talagrand inequality yields the same conclusion.
Theorem 3.12. Assume that (M,d) is a metric space and let h >0 be given. If m is a probability measure on (M,d) that satisfies an h-Talagrand inequality of constant K > 0 then all Lipschitz functions are exponentially integrable. More precisely, for any Lipschitz function ϕ with kϕkLip ≤ 1 and R ϕdm= 0 we have
∀t >0, Z
M
etϕdm≤et
2 2K+ht, or equivalently, for any Lipschitz function ϕ
∀t >0, Z
M
etϕdm≤exp
t Z
M
ϕdm
exp t2
2Kkϕk2Lip+htkϕkLip
. An independent, alternative approach to generalized Ricci curvature bounds for discrete spaces, based on optimal transportation as well, was pre- sented by Yann Ollivier [16]. In the continuous case, the following two asser- tions are equivalent (see [18] for the case of Riemannian manifolds):
(i) The entropy functional Ent(·|m) is weaklyK-convex on P2(M,d), in the sense of inequality (2.1).
