Asymptotic isoperimetry of balls in metric measure spaces

In this paper, we study the asymptotic behavior of the volume of spheres in metric measure spaces. We first introduce a general setting adapted to the study of asymptotic isoperimetry in a general class of metric measure spaces. We then introduce a notion of"being asymptotically isoperimetric"for a family of finite and unbounded measure subsets. We show that there exist graphs with uniform polynomial growth whose balls are not asymptotically isoperimetric and we discuss the stability of related properties under quasi-isometries. Finally, we study the asymptotically isoperimetric properties of connected subsets in a metric measure space. In particular, we build graphs with uniform polynomial growth whose connected subsets are not asymptotically isoperimetric.


Boundary of a subset and isoperimetric profile.
Let (X, d, µ) be a metric measure space. Let us denote by B(x, r) the open ball of center x and radius r. We suppose that the measure µ is Borel and σ-finite. For any measurable subset A of X, any h > 0, write: A h = {x ∈ X, d(x, A) ≤ h}, and: Let us call ∂ h A the h-boundary of A, and ∂ h B(x, r) the h-sphere of center x and radius r. Definition 1.1. Let us call the h-profile the nondecreasing function defined on R + by: where A ranges over all µ-measurable subsets of X with finite measure.
This definition of large-scale boundary has the following advantage: under some weak properties on the metric measure space X, we will see in Section 2.1 that in some sense, the boundary of a subset A ⊂ X has a thickness "uniformly comparable to h". This will be play a crucial role in the proof of the invariance of "asymptotic isoperimetric properties" under large-scale equivalence.
We could also define the boundary of a subset A by ∂ h A = A h A. But with this definition, the thickness of the boundary may have uncontrollable "fluctuations". Indeed, consider a Riemannian manifold X with boundary and take a subset A which is at distance ε > 0 from the boundary of X. Assume that ε ≪ h. With the latter definition, the thickness of ∂ h A may vary between h and ε and its volume may strongly depend on ε, even if X has bounded geometry. Note that this problem disappears with our definition since every point of the boundary of A is close to a ball of radius h/2 included in ∂ h A (see Section 2.1 for precise statements).

Lower/upper profile restricted to a family of subsets.
Let (X, d, µ) be a metric measure space. In order to study isoperimetric properties of a family of subsets of X with finite, unbounded volume, it is useful to introduce the following notions: Definition 1.2. Let A be a family of subsets of X with finite, unbounded volume. We call lower (resp. upper) h-profile restricted to A the nondecreasing function I ↓ h,A defined by: (resp. I ↑ h,A (t) = sup µ(A)≤t,A∈A µ(∂ h A)). Definition 1.3. Consider two monotone functions f and g: R + → R + . Say that f ≈ g if there exist some constants C i such that C 1 f (C 2 t) ≤ g(t) ≤ C 3 f (C 4 t) for all t ∈ R + .
The asymptotic behavior of a monotone function R + → R + may be defined as its equivalence class modulo ≈.
We get a natural order relation on the set of equivalence classes modulo ≈ of monotone functions defined on R + by setting: We say that the family A is asymptotically isoperimetric (resp. strongly asymptotically isoperimetric) if for all A ∈ A Remark 1.4. Note that asymptotically isoperimetric means that for any t we can always choose an optimal set among those of A whose measure is larger than t whereas strongly asymptotically isoperimetric means that every set of A is optimal (but the family (µ(A)) A∈A may be lacunar). In almost all cases we will consider, the family (µ(A)) A∈A will not be lacunar, and strong asymptotic isoperimetry will imply asymptotic isoperimetry.

Large scale study.
Let us recall the definition of a quasi-isometry (which is also sometimes called rough isometry).
Definition 1.5. Let (X, d) and (X ′ , d ′ ) two metric spaces. One says that X and X ′ are quasiisometric if there is a function f from X to X ′ with the following properties.
(a) there exists C 1 > 0 such that [f (X)] C1 = X ′ . (b) there exists C 2 ≥ 1 such that, for all x, y ∈ X, , f (y)) ≤ C 2 d(x, y) + C 2 . Example 1.6. Let G be a finitely generated group and let S 1 and S 2 two finite symmetric generating sets of G. Then it is very simple to see that the identity map: G → G induces a quasi-isometry between the Cayley graphs (G, S 1 ) and (G, S 2 ). At the beginning of the 80's, M. Gromov (see [5]) initiated the study of finitely generated groups up to quasi-isometry.
Example 1.7. The universal cover of a compact Riemannian manifold is quasi-isometric to every Cayley graph of the covering group (see [5] and [11]).
Note that the notion of quasi-isometry is purely metric. So, when we look for quasi-isometry invariant properties of a metric measure space like, for instance, volume growth, we are led to assume some uniformity properties on the volume of balls. This is the reason why, for instance, this notion is well adapted to geometric group theory. But since we want to deal with more general spaces, we will define a more restrictive class of maps. Those maps will be asked to preserve locally the volume of balls. On the other hand, we want local properties to be stable under bounded fluctuations of the metric. Precisely, let (X, d, µ) be a metric measure space and let d ′ be another metric on X such that d/d ′ and d ′ /d are bounded. The following definition (see [1]) prevents wild changes of the volume of balls with bounded radii under the identity map between (X, d, µ) and (X, d ′ , µ). Definition 1.8. Let us say that (X, d, µ) is doubling at fixed radius, or has property (DV ) loc if for all r > 0, there exists C r > 0 such that, for all x ∈ X: µ(B(x, 2r)) ≤ C r µ(B(x, r)). Remark 1.9. Note that Property (DV ) loc is local in r but uniform in x. Example 1.10. Bounded degree graphs or Riemanniann manifolds with Ricci curvature bounded from below satisfy (DV ) loc .
The following notion was introduced by Kanai [7] (see also [1]). Definition 1.11. Let (X, d, µ) and (X ′ , d ′ , µ ′ ) two metric measure spaces with property (DV ) loc . Let us say that X and X ′ are large scale equivalent (we can easily check that it is an equivalence relation) if there is a function f from X to X ′ with the following properties: there exist some constants C 1 > 0, C 2 ≥ 1, C 3 ≥ 1 such that: (a) f is a quasi-isometry of constants C 1 and C 2 ; (b) for all x ∈ X: Focusing our attention on balls of radius 1 may not seem very natural. Nevertheless, this is not a serious issue since property (DV ) loc allows to make no distinction between balls of radius 1 and balls of radius C for any constant C > 0. Remark 1.12. Note that for graphs with bounded degree (equipped with the counting measure), or Riemannian manifolds with bounded Ricci curvature (equipped with the Riemannian measure), quasi-isometries are automatically large-scale equivalences.

Volume of balls and growth function.
Let (X, d, µ) be a metric measure space. The equivalence class modulo ≈ of µ(B(x, r)) is independent from x. We call it the volume growth of X and we write it V (r). One can easily show the next result (see [1]).
Proposition 1.13. The volume growth is invariant under large-scale equivalence (among (DV ) loc spaces).
Definition 1.14. Let X be a metric measure space. We say that X is doubling if there exists a constant C > 0 such that, ∀x ∈ X and ∀r ≥ 0: We will call this property (DV ).
Remark 1.15. It is easy to see that (DV ) is invariant under large scale equivalence between (DV ) loc spaces. To be more general, we could define an asymptotic doubling condition (DV ) ∞ , restricting (1.1) to balls of radius more than a constant (depending on the space). Property (DV ) ∞ is also stable under large-scale equivalence between (DV ) loc spaces and has the advantage to focus on large scale properties only. Actually, in every situation met in this paper, the assumption (DV ) can be replaced by (DV ) ∞ + (DV ) loc (note that they are equivalent for graphs). Nevertheless, for the sake of simplicity, we will leave this generalization aside.
Example 1.16. A crucial class of doubling spaces is the class of spaces with polynomial growth: we say that a metric measure space has (strict) polynomial growth of degree d if there exists a constant C ≥ 1 such that, ∀x ∈ X and ∀r ≥ 1: Note that Gromov (see [4]) proved that every doubling finitely generated group is actually of polynomial growth of integer degree (since it is virtually nilpotent).

Contents of the article.
In the next section, we present a setting adapted to the study of asymptotic isoperimetry in general metric measure spaces. The main interest of this setting is that the "asymptotic isoperimetric properties" are invariant under large-scale equivalence. In particular, it will imply that if X is a (DV ) loc and uniformly connected (see next section) space, then the class modulo ≈ of I h will not depend any more on h provided h is large enough. For that reason, we will simply denote I instead of I h . Then, we introduce a notion of weak geodesicity (see definition 2.15) which is invariant under Hausdorff equivalence but not under quasi-isometry. We call it property (M) since it can be formulated in terms of existence of some "monotone" geodesic chains between any pair of points. This property plays a crucial role when we want to obtain upper bounds for the volume of spheres (see [12]). It will also appear as a natural condition for some properties discussed in this paper.
Here are the two main problems concerning isoperimetry in metric measure spaces: first, determining the asymptotic behavior of the profile; second, finding families of subsets that optimize the profile. The asymptotic behavior of I is more or less related to volume growth (see [2] and [8] for the case of finitely generated groups). In the setting of groups, the two problems have been solved for Lie groups (and for polycyclic groups) in [9] and [2] and for a wide class of groups constructed by wreath products in [3]. It seems very difficult (and probably desperate) to get general statements for graphs with bounded degree without any regularity assumption (like doubling property or homogeneity). On the other hand, let us emphasize the fact that doubling condition appears as a crucial assumption in many fields of analysis. So in this article, we will deal essentially with doubling metric measure spaces. Without any specific assumption on the space, balls seem to be natural candidates for being isoperimetric subsets, especially when the space is doubling (see Corollary 3.4).
One could naively think that thanks to Theorem 2.10, properties like asymptotic isoperimetry of balls are stable under large-scale equivalence. Unfortunately, it is not the case. Indeed, Theorem 2.10 roughly says that if f : X → X ′ is a large scale equivalence between two metric measured spaces, then for a > 0 large enough and for every measurable subset A of X, the measure of the boundary of [f (A)] a is smaller than the measure of the boundary of A up to a multiplicative constant. So, in order to apply Theorem 2.10 to asymptotic isoperimetry of balls, we need the existence of some C > 0 such that: This condition is satisfied if f is a Hausdorff equivalence. But if f is a quasi-isometry, we only have: I.
Otherwise, we will say that X is (NIB). • We say that X is strongly-(IB) if balls are strongly asymptotically isoperimetric, i.e. if I ↑ B I.
• Finally, we say that a metric measure space is stably-(IB) (resp. stably-(NIB)) if every (M)-space (see Definition 2.15) isometric at infinity to X is (IB) (resp. (NIB)). If necessary, we will restrict our study to a certain class of metric measure spaces.
Definition 1.18. We say that a space (X, d, µ) satisfies a strong (isoperimetric) inequality-or that X has a strong profile-if I id/φ where φ is the equivalence class modulo ≈ of the function: t → inf{r, µ(B(x, r) ≥ t}.
We will show that every doubling space satisfying a strong isoperimetric inequality satisfies (IB). This actually implies that such a space satisfies stably-(IB). In particular, any compactly generated, locally compact group of polynomial growth satisfies (IB). In contrast, apart from the Abelian case [12], it is still unknown whether such a group G satisfies strongly-(IB) or not, or, in other words, if we have µ(K n+1 K n ) ≈ n d−1 where K is a compact generating set of G and µ is a Haar measure on G.
Conversely, we will show that every strongly-(IB) doubling space satisfies a strong isoperimetric inequality. On the other hand, we will see that the strong isoperimetric inequality does not imply strongly-(IB), even if the volume growth is linear (V (r) ≈ r).
To see that strongly-(IB) is not stable under large scale equivalence, even among graphs with polynomial growth, we shall construct a graph quasi-isometric to Z 2 whose volume of spheres is not dominated by r log 3/ log 2 (where r is the radius). Note that this can be compared with the following result (see [12], theorem 1) : Theorem 1.19. [12] Let X be a metric measure space with properties (M) and (DV ) (for instance, a graph or a complete Riemannian manifold with the doubling property). There exists δ > 0 and a constant C > 0 such that, ∀x ∈ X and ∀r > 0: In particular, the ratio µ(∂B x,r (x))/µ(B(x, r)) tends to 0 uniformly in x when r goes to infinity.
When the profile is not strong, we will see that a many situations can happen. All the counterexamples built in the corresponding section will be graphs of polynomial growth.
The case of a bounded profile is quite specific. Indeed, in that case, and under some hypothesis on X (including graphs and manifolds with bounded geometry), we will prove that if (P n ) n∈N is an asymptotically isoperimetric sequence of connected subsets of X, one can find a constant C ≥ 1 and ∀n ∈ N, some x n ∈ X, r n > 0 such that: Note that here, we don't ask X to be doubling.
Nevertheless, we will see that there exist graphs with polynomial growth (with unbounded profile) such that no asymptotically isoperimetric family has this property. In particular, those graphs are stably-(NIB).
To be complete, we also build graphs with polynomial growth, bounded profile and satisfying stably-(NIB).
Concerning the stability under large-scale equivalence, we will see that even among graphs with polynomial growth, with bounded or unbounded profile, property (IB) is not stable under large-scale equivalence (in the case of graphs equipped with the counting measure, a large-scale equivalence is simply a quasi-isometry).
Finally, we shall examine isoperimetric properties of connected subsets (say that A is (metrically) connected if for any partition A = A 1 ⊔A 2 , with d(A 1 , A 2 ) ≥ 10, either A 1 or A 2 is empty). Clearly, since balls of a (M)-space are connected, the strong isoperimetric inequality implies that connected sets are asymptotically isoperimetric. On the other hand, we will show that there exist graphs with polynomial growth whose connected subsets are not asymptotically isoperimetric.
2. Isoperimetry at infinity: a general setting 2.1. Isoperimetric at a given scale. The purpose of this section is to find some minimal conditions under which "isoperimetric properties at infinity" become invariant under large-scale equivalence. In the introduction, namely in Section 1.1.1, we justified our definition of the boundary by the fact that we want it to have a uniform thickness. Nevertheless, it is not suffisant to our purpose: it is also important for the space X to look connected at scale h. Let X be a graph; if h = 1/2, then every subset of X has a trivial boundary, so that all the isoperimetric properties of X are trivial.
Definition 2.1. Let X be a metric space and fix b > 0. Let us call a b-chain of length n from x to y, a finite sequence The following definition can be used to study the isoperimetry at a given scale, although we will only use it "large-scale version" in this paper.
then we say that X is uniformly b-connected (or merely uniformly connected). Remark 2.3. Note that in the scaled version, the space X is allowed to have a proper nonempty Invariance under quasi-isometry: Note that if X is uniformly b-connected at scale ≤ E 1 and if f : X → X ′ is a quasi-isometry of constants C 1 and C 2 smaller enough than E, then X ′ is uniformly Remark 2.4. Let us write d b (x, y) for the b-distance from x to y, that is, the minimal length of a b-chain between x and y (note that if every couple of points of X can be joined by a b-chain, then d b is a pseudo-metric on X).
If there exists C > 0 such that, for all x, y ∈ X, one has: d b (x, y) ≤ Cd(x, y) + C, then in particular, X is uniformly b-connected (we can call this property: quasi-geodesic property).
Example 2.5. A graph and a Riemannian manifold are respectively uniformly 1-connected and uniformly b-connected for all b > 0.
(ii) If, moreover, X is (DV ) loc and h ≪ E 1 , then there exists a constant C ′ ≥ 1 such that, for every subset A, there exists a family (B(y i , b)) i included in ∂ h A, such that, for all i = j, d(y i , y j ) ≥ E 2 and such that: (iii) The h-boundary measure of a subset of a (DV ) loc , uniformly b-connected space does not depend on h up to a multiplicative constant, provided E 1 ≫ h ≥ 2b.
Proof : Let x ∈ A c such that d(x, A) < E 1 and let y ∈ A be such that d(x, y) ≤ E 1 . We know from the hypothesis that there exists a finite chain x 0 = x, x 1 , . . . , x n = y satisfying: Let us show the second assertion. Consider a maximal family of disjoint balls (B( Using the first assertion and the fact that h ≪ E 1 , one sees that each B(x i , 2E 2 ) contains a ball B(y i , b) included in ∂ h A. It is clear that the balls B(y i , 10E 2 ) form a covering of ∂ h A and that the balls (B(y i , b) are disjoint. But, by property (DV ) loc , there exists C ′ ≥ 1, depending on b and E 2 , such that, for all i ∈ I: which proves (ii). The assertion (iii) now follows from (ii).
Remark 2.7. This proposition gives conditions to study isoperimetry at scale between b and E 1 , i.e. choosing h far from those two bounds. Thus, we will always assume that this condition holds and we will simply write ∂A instead of ∂ h A. Otherwise, problems may happen. We talked about what can occur if h < b at the beginning of this section. Now, let us give an idea of what can happen if h > E 1 . Consider a metric measure space X such that X = ∪ i∈I X i where the X i are subsets such that d(X i , X j ) ≥ E 1 whenever i = j and such that µ(X i ) is finite for every i ∈ I but not bounded. Note that for h < E 1 the boundary of every X i is empty so that the family (X i ) i∈I is trivially asymptotically isoperimetric. But this can change dramatically if h > E 1 because the boundary of X i can meet many X j 's for j = i.
Remark 2.8. If we replace uniformly b-connected at scale ≤ E 1 by uniformly b-connected, then the proposition gives a setting adapted to the study of large scale isoperimetry. Namely, it says that for a uniformly b-connected, (DV ) loc space, the choice of h does not matter, provided h ≥ 2b.
So, from now on, we will simply call "profile" (instead of h-profile) the equivalence class modulo ≈ of I h . Note that the same holds for restricted profiles I ↓ h,A , and I ↑ h,A that we will simply denote I ↓ A and I ↑ A (where A is a family of subsets of X). The following theorem shows that a large-scale equivalence f with controlled constants essentially preserves all isoperimetric properties.
. We suppose also that E 1 and E ′ 1 are far larger than C 1 , C 2 , C 2 b and C 2 (b ′ + C 1 ). Then, there exists a constant K ≥ 1 such that, for any subset A of finite measure: Proof : Let us start with a lemma: Lemma 2.11. Let X be a (DV ) loc space and fix some α > 0. Then there exists a constant c > 0 such that, for all family (B(x i , α)) i∈I of disjoint balls of X, there is a subset J of I such that ∀j ∈ J, the balls B(x j , 2α) are still disjoint, and such that: Proof : Let us consider a maximal subset J of I such that (B(x j , 2α)) j∈J forms a family of disjoint balls. Then, by maximality, we get: We conclude thanks to property (DV ) loc .
To fix ideas, take h = 2b and h ′ = 2b ′ . Assertion (ii) of Proposition 2.6 implies that there exists a family of balls (B(y By the lemma, and up to changing the constant C ′ , one can even suppose that d( The points x i are then at distance ≫ E 2 to one another. Moreover, since y i is both at distance ≤ 2b + C 1 of f (A) and of f (A c ), x i is both at distance ≪ E 1 of A and of A c . So, by the assertion (i) of the proposition, there exists The theorem then follows from property (DV ) loc and from property of "almost-conservation" of the volume (property (b)) of large-scale equivalence.
Remark 2.12. Note that in the case of graphs, the condition h ≥ 2 can be relaxed to h ≥ 1 (the proposition and the theorem stay true and their proofs are unchanged).
Corollary 2.13. Under the hypotheses of the theorem, we have: (ii) if I and I ′ are the profiles of X and X ′ respectively, we get: The corollary results immediately from the theorem and the following proposition.
Proposition 2.14. Let f be a large-scale equivalence between two (DV ) loc spaces X and X ′ . Then for every subset A of X, there exists C ≥ 1 such that: Proof : Consider a maximal family of disjoint balls (B(y i , C 1 )) i∈I whose centers belong to f (A).
By property (DV ) loc and property of almost-conservation of the measure of small balls (property (b)) of f , the measure of this ball is comparable to that of B(y i , 3C 1 ). So we are done.
Finally, let us mention that if we suppose that X and X ′ are uniformly connected and satisfy the (DV ) loc condition, then Theorem 2.10 and its corollary hold for any large-scale equivalence f .

Property (M): monotone geodesicity.
Let us introduce a natural (but quite strong) property of geodesicity.
Remark 2.16. Let (X, d) be a (M) metric space. Then X has "monotone geodesics" (this is why we call this property (M)): i.e. there exists C ≥ 1 such that, for all x, y ∈ X, there exists a finite chain x 0 = x, x 1 , . . . , x n = y such that ∀0 ≤ i < n, Consequently, ∀r, k > 0, ∀y ∈ B(x, r + k), we have: These two properties are actually trivially equivalent to property (M).
Recall (see [6], p 2) that two metric spaces X and Y are said Hausdorff equivalent: , which is invariant under quasi-isometry: X is quasi-geodesic if there exist two constants d > 0 and λ > 0 such that for all (x, y) ∈ X 2 there exists a finite chain of points of X: Example 2.18. A geodesic space has property (M), in particular graphs and complete Riemannian manifolds have property (M). A discretisation (i.e. a discrete net) of a Riemannian manifold X has property (M) for the induced distance.
Remark 2.19. Note that in general, if X is a metric measure space, we have: Moreover, if X has property (M), then, we have: Note that this is not true in general, even for quasi-geodesic spaces.
3. Link between isoperimetry of balls and strong isoperimetric inequality

Strong isoperimetric inequality implies (IB).
To fix ideas, spaces we will consider from now on will be (DV ) loc and uniformly 1-connected. Let us write: ∂A = ∂ 2 A for any subset A of a metric space X.
Let X be a metric measure space. Let V be a nondecreasing function belonging to the volume growth class (for instance: V (r) = µ(B(x, r)) for a x ∈ X). Write φ(t) = inf{r, V (r) ≥ t} for the "right inverse" function of V . Remark that if f and g are nondecreasing functions R + → R + , then f ≈ g if and only if their right inverses are equivalent. In particular, the equivalence class of φ is invariant under large-scale equivalence.
Remark that this is equivalent to: I id/φ, Therefore, if X satisfies a strong isoperimetric inequality, we will say that it has a strong profile.
Example 3.2. If X has polynomial growth of degree d, we have: φ(t) ≈ t 1/d . So X has a strong profile if and only if: Write, for all x ∈ X and for all 0 < r < r ′ : ∀x ∈ X, ∀r ≥ 1, inf Proof : Clearly, it suffices to prove the proposition when r = n is a positive integer. First, note that ∪ 2n We conclude by Doubling property.
Corollary 3.4. Let X be a uniformly connected doubling space. Then we have: Namely, there exists a constant C ≥ 1 such that: ∀x ∈ X, ∀r > 0, inf Proof. This follows from Remark 2.19.
Corollary 3.5. Let X be a uniformly connected doubling space satisfying a strong isoperimetric inequality. Then, X is stably-(IB).
Proof : It follows from Corollary 3.4 and from Corollary 2.13.
Remark 3.6. Varopoulos [13] showed that the strong isoperimetric inequality is satisfied by any group of polynomial growth. Coulhon and Saloff-Coste [2] then proved it for any unimodular compactly generated locally compact group with a simple and elegant demonstration. We have the following corollary.

The strong isoperimetric inequality does not imply strongly-(IB).
Note that this will result from the example shown in section 3.3. Let us present here a counterexample with linear growth.
For every integer n, we consider the following finite rooted tree G n : first take the standard binary tree of depth n. Then stretch it as follows: replace each edge connecting a k − 1'th generation vertex to a k'th generation vertex by a (graph) interval of length 2 2 n−k . Then consider the graph G ′ n obtained by taking two copies of G n and identifying the vertices of last generation of the first copy with those of the second copy. Write r n and r ′ n for the two vertices of G ′ n corresponding to the respective roots of the two copies of G n . Finally, glue "linearly" the G ′ n together identifying r ′ n with r n+1 , for all n: it defines a graph X.
Let us show that X has linear growth (i.e. polynomial growth of degree 1). Thus I ≈ 1, and since the boundary volume of balls is clearly not bounded, we do not have I ↑ B I. In particular, X is not strongly-(IB).
Since X is infinite, it is enough to show that there exists a constant C > 0 such that for every vertex x of X. But it is clear that among the balls of radius r, those which are centered in points of n'th generation of a G n for n large enough are of maximal volume. Let us take such an x. Remark that for k j=0 2 2 j ≤ r ≤ k+1 j=0 2 2 j , we have: So it is enough to show (3.1) for r = k j=0 2 2 j . We have: Which proves (3.1) with C = 8.
Remark 3.8. This example and that of section 3.3 show in particular that the strong isoperimetric inequality does not imply (even in linear growth case) strongly-(IB).

Instability of strongly-(IB) under quasi-isometry.
Theorem 3.9. We can find a graph, quasi-isometric to Z 2 (resp. a Riemannian manifold M bi-Lipschitz equivalent to R 2 ) whose volume of spheres is not dominated by r log 3/ log 2 (where r is the radius).
Remark 3.10. The restriction to dimension 2 is not essential, but was made to simplify the exposition (actually, we merely need the dimension to be greater or equal to 2).
Proof : The general idea of the construction is to get a sequence of spheres which look like finitely iterated Von Koch curves. First, we will build a graph with weighted edges. Actually, this graph will be simply the standard Cayley graph of Z 2 , and the edges will have lengths equal to 1 except for some selected edges which will have length equal to a small, but fixed positive number. First step of the construction: Let us define a sequence (A k ) of disjoint subtrees of Z 2 (which is identified to its usual Cayley graph). Let (e 1 , e 2 ) be the canonical basis of Z 2 and denote S = {±e 1 , ±e 2 }. For every k ≥ 1, let a k = (2 2k , 0) be the root of the tree A k and define A k by: It is easy to see that A k is a subtree of Z 2 and that the above decomposition of x is unique. In particular, we can consider its intrinsic graph metric d A k : let S k be the sphere of center a k and of radius 2 k+1 − 1 for this metric. Clearly, |S k | ≥ 3 k−1 . Second step of the construction: We define a graph Y with weighted edges as follows: Y is the usual Cayley graph of Z 2 ; all edges of Y have length 1 but those belonging to A = ∪ k A k which have length equal to 1/100. The measure on Y is the countable measure and the distance between two vertices v and w is the minimal length of a chain joining v to w, the length of a chain being the sum of the weights of its edges. Clearly, as a metric measure space, Y is large-scale equivalent to Z 2 .
For every k ≥ 2, consider the sphere S(a k , r k ) = B(a k , r k + 1) B(a k , r k ) of Y , where r k = (2 k+1 − 1)/100. Claim 3.11. We have: S k ⊂ S(a k , r k ), so that: Proof : Note that the claim looks almost obvious on a drawing. Nevertheless, for the sake of completeness, we give a combinatorial proof. Let us show that a geodesic chain in the tree A k is also a minimizing geodesic chain in Y . Applying this to a geodesic chain between a k and any element of S k (which is of length r k in Y ), we have that S k ⊂ S(a k , r k ), so we are done.
So let x be a vertex of A k . By (3.2), we have: Let us show by recurrence on d Y (a k , x) (which takes discrete values) that 100 If x = a k , there is nothing to prove. Consider c = (c(0) = x, c(2), . . . , c(m) = a k ) a minimal geodesic chain in Y between a k and x. Clearly, it suffices to prove that c ⊂ A k . Suppose the contrary. Let t be the largest positive integer such that c(t) belongs to A k and c(t + 1) does not. Let l be the smallest positive integer such that c(t + l) ∈ A k , so that (c(t + 1), . . . , c(t + l − 1)) is entirely outside of A k . By recurrence, the chain (c(t + l), . . . , c(m)) is in A k . Thus we have: Since c is a minimal chain, we also have: The following lemma applied to u = c(t) and v = c(t + l) implies that t = t + l which is absurd since it means that c is included in A k . Lemma 3.12. let u and v be in A k . We have: Note that by construction, and since A k is a tree, On the other hand, we have: First, assume that u 1 = v 1 . Then, by (3.2), the projection of u 1 − v 1 along e 1 or e 2 is not zero and belongs to 2 k−i(v) N. Moreover, using the fact that ε j+1 (u) = −ε j (u)) for every j, the same projection of So we are done. Now, assume that a k )).
Clearly, Y is quasi-isometric to Z 2 . It is not difficult (and left to the reader) to see that we can adapt the construction to obtain a graph. Now, let us explain briefly how we can adapt the construction to obtain a Riemannian manifold bi-Lipschitz equivalent to R 2 . First, we embed Z 2 into R 2 in the standard way, so that A k is now a subtree of R 2 . LetÃ be the 1/100-neighborhood of A in R 2 . Let f be a nonnegative function defined on R 2 such that: 1 − f is supported byÃ, f ≥ a and f (x) = a for all x ∈ A. Finally, define a new metric on R 2 multiplying the Euclidian one by f .

Strongly-(IB) implies the strong isoperimetric inequality.
The converse to Proposition 3.5 is clearly false (see the examples of the next section). However, one has: Proposition 3.13. Let X be a doubling (M)-space. Suppose moreover that there exists x ∈ X such that the family of balls of center x is strongly asymptotically isoperimetric. Then we have In particular, X satisfies a strong isoperimetric inequality.
Proof. Since (B(x, r)) r forms an asymptotically isoperimetric family, it is enough to show that there exists c > 0 such that: µ(∂B(x, r)) ≥ c µ(B(x, r)) r .
So we are done.

What can happen if the profile is not strong
All the metric measure spaces built in this section will be graphs with polynomial growth. For simplicity, we write |A| for the cardinal of a finite subset A of a graph.

4.1.
Bounded profile: connected isoperimetric sets are "controlled" by balls. We will say that a subset A of a metric space is metrically connected (we will merely say "connected" from now on) if there does not exist any nontrivial partition of Let X be a uniformly 1/2-connected space, with bounded profile, and such that balls measure of radius 1/2 is more than a constant a > 0. Actually, we can ignore nonconnected sets. Indeed if (A n ) is an isoperimetric family, then the A n have a bounded number of connected components: otherwise, by Proposition 1.13, the boundary of A n would not be bounded (because the distinct connected components have disjoint 1-boundaries each one containing a ball of radius 1/2). It is enough to replace A n by its connected component of maximal volume. Claim 4.1. Let (X, d, µ) be a (DV ) loc , uniformly 1/2-connected space such that the measure of balls of radius 1/2 is more than a > 0 and whose profile I is bounded. Then, if (A n ) is an isoperimetric sequence of connected subsets of X, there exist a constant C > 0, some x n ∈ X and some r n > 0 such that: ∀n, B(x n , r n ) ⊂ A n ⊂ B(x n , Cr n ).
Proof : To fix ideas, let us assume that ∂A = ∂ 1 A (for all A ⊂ X). Let y n be a point of A n and write d n = sup y∈∂An d(y n , y). Let r ≤ d n be such that C r,r+1 (y n ) intersects nontrivially ∂A n (recall that C r,r ′ (x) = B(x, r ′ ) \ B(x, r)). Then, by Proposition 2.6, there exists a constant C ≥ 1 such that C r−C,r+C (y n ) ∩ ∂A n contains a ball of radius 1/2 and therefore has measure ≥ a. Consequently, if δ n = sup{r ′ − r; C r,r ′ (y n ) ∩ ∂A n = ∅}, then: Since the boundary of A n has bounded measure, there exists a constant c > 0 and, for all n, two positive reals r ′ n and r" n such that r" n − r ′ n ≥ cd n and C r ′ n ,r"n ∩ ∂A = ∅. Write s n = (r ′ n + r" n )/2. Since A n is connected, C sn−10,sn+10 (x) ∩ A n is nonempty. But then, if x n ∈ C sn−10,sn+10 (x) ∩ A n , we get: On the other hand: Write: r n = cd n /2 − 10. The proposition follows from (4.1) and from (4.2).

4.2.
Stably-(NIB) graphs with unbounded profile and where isoperimetric families can never be "controlled" by families of balls.
Theorem 4.2. For every integer d ≥ 2, there exists a graph X of polynomial growth of degree d, with unbounded profile, satisfying stably-(NIB) and such that, for all isoperimetric sequences (A n ), it is impossible to find sequences of balls B n = B(x n , r n ) and B ′ n = B(x ′ n , r ′ n ) of comparable radii (i.e. such that r ′ n /r n is bounded) such that: B n ⊂ A n ⊂ B ′ n , ∀n. Consider the graph X obtained from Z d taking off some edges. Consider, in the axis Z.e 1 , the intervals (S n ) of length [ √ n] and at distance 2 n from one another. Consider the sequence (A n ) of full parallelepiped defined by the equations: x 1 ∈ I n and |x i | ≤ n/2 for i ≥ 2.
Then consider a partition of the boundary (in Z d ) of A n in (d − 1)-dimensional cubes a k n whose edges have length approximatively √ n. Remove all the edges that connect A n to its complementary but those connected to the "center" of a k n (here, the center of a k n is a point of Z d we choose at distance ≤ 2 from the "true center" in R n of the convex hull of a k n ). We thus obtain a connected graph X. Note that the A n are such that |A n | ≈ n d−1 √ n and: Write A for the union of A i and A c for its complementary in X.
Claim 4.3. The growth in X is polynomial of degree d.
Proof : It will follow from the strong profile of balls.
Claim 4.5. Let R be a unbounded subset of R + and let (P r ) r∈R be a family of subsets such that there exist two constants C ≥ 1 and a > 0 such that: Then there exists a constant c ′ such that: The following lemma and its proof will be useful in all examples that we will expose in the following sections. Write A c for the complementary of A (in X or, which is actually the same in Z d ).
Lemma 4.6. The profile of A c (or of A ′c ) is strong. That means: Proof of the lemma. First of all, it is enough to consider only connected subsets P of A c . Indeed, if P has many connected components P 1 . . . P k , then, by subadditivity of the function φ : t → t d−1 d , if the P i verify: |∂P i | ≥ cφ(|P i |), then so do P .
Note that A c embeds into X and into Z d . The idea consists in comparing the profile of A c to that of Z d . First of all, let us assume that a connected subset P of A c -seen in X-intersects the boundary of many A n . Then, as |A n | is negligible compared to the distance between the A n when n goes to infinity, the set of points of ∂ Z d P at distance 1 of A has negligible volume compared to |∂P |. Thus, if |P | et n are large enough, we get: So it is enough to consider subsets meeting only one A n . But the complementary of a convex polyhedra of Z d has trivially the same profile (up to a constant) as Z d . So we are done.
Proof of the claim 4.5. Let (P r ) be a family of subsets of X satisfying the condition of the proposition. We have to show that ∀r, |∂P r | ≥ c ′ |P r | d−1 d . If P ⊂ A c , the claim is a direct consequence of the lemma.
Suppose that P meets some A n and that r ≥ 100C √ n. Then we have already seen (in the proof of Lemma 4.2) that if many A n intersect P r , the cardinal of the intersection of this P r with A are negligible compared to its boundary provided n and |P r | are large enough. We can thus suppose that P r meets only one A n . Furthermore, since r ≥ 100 (n), there is some x ′ in B(x r , r/C) such that B(x ′ , r/10C) ∈ B(x r , r/C) ∩ A c . Then, observe that since B(x ′ , r/10C) ⊂ [P r ] a , there is a constant c > 0 such that: It follows that the intersection of P r with A c has volume ≥ c ′ |P r | wherec is a constant depending only on C and a. So by Lemma 4.2, we have: We then have to study the case r ≤ 100C √ n. We can assume that x r ∈ A n (otherwise, we conclude with Lemma 4.2). Let π be the orthogonal projection on the hyperplane x 2 = 0. Then for n large enough, Cr is smaller than n/2. Consequently, since P r ∈ B(x r , Cr), every point of π(P r ) has at least one antecedent in the boundary of P r . So, we have: Moreover, note that π(B(x r , r/C)) = B(π(x r ), r/C) (note that this ball lies in Z d−1 ). On the other hand, since the projection is 1-Lipschitz, we get: B(π(x r ), r/C) ⊂ [π(P r )] a . Similarly to (4.3), we have: |π(P r ) ∩ B(π(x r ), r/C| ≥ c|B(π(x r ), r/C)| So, finally, we have: so we are done.
Corollary 4.7. In every space isometric at infinity to X, the volume of spheres ≈ r d−1 . In particular, they are not asymptotically isoperimetric.
Proof of the corollary. Let f : X ′ → X a large-scale equivalence between two metric measure spaces X ′ and X and take y ∈ X ′ . It comes: The corollary follows from Claim 4.5 and from Theorem 2.10.

4.3.
Graphs stably-(NIB) with bounded profile. The construction follows the same lines as in the previous section. Consider in Z d , a sequence (C n ) of subsets defined by: We disconnect C n from the rest everywhere but in the axis Z.e 1 . Let Y be the corresponding graph. C n looks like a ball (of Z d ) "constricted" at the equator. Indeed, every point of C n belonging to the hyperplane {x 2 = 0} is at distance at most log n from the boundary (in Y ) of C n . This is the property that will prevent C n from being "deformed" into a ball. Write C = ∪ n C n . Lemma 4.9. The graph C c has a strong profile.
Proof : The demonstration is essentially the same as for Lemma 4.2.
Claim 4.10. The growth in the graph X is polynomial of degree d.
Proof : We have to show that there exists a constant c > 0 such that, ∀x, r, |B(x, r)| ≥ cr d (the converse inequality following from the fact that X embeds in Z d ). Thanks to Lemma 4.9, we can suppose that B is included in a C n0 so that its radius is ≤ n 0 .
The conclusion follows then from the next trivial fact: in Z d , if r ≤ n 0 , the volume of the intersection of a ball of radius n 0 with a ball of radius r ≤ n 0 and of center belonging to the first ball is ≥ 2 −d |B(x, r)| ≥ 2 −10d r d . Indeed, the worst case is when x is in a "corner" of the ball. So we are done.
Claim 4.11. If Y ′ is a (M)-space which is isometric at the infinity to Y , then its balls are not asymptotically isoperimetric.
Proof : The demonstration results from the following lemma and Proposition 1.13. Lemma 4.12. Let P be an asymptotically isoperimetric family of connected subsets of X. Then there exists a constant C ≥ 1 such that, for all P ∈ P of measure > C, there exists n such that |P C n | ≤ C.
Proof : Since the profile of C c is strong, it is clear that for |P | large enough, P ∩ C c must be bounded. We then have to show that if (P n ) is a sequence of subsets such that for all n, P n ⊂ C n and such that |P n | and |C n \ P n | tends to infinity, then |∂P n | also tends to infinity. Suppose, for instance that |P n | ≤ |C n \ P n |. But Theorem 2.10 makes clear that this problem in Z d is equivalent to the similar problem in R d : that is, replacing C n with its convex hullC n in R d . Since theC n are homothetic copies ofC 1 , by homogeneity, we only have to show that the profile I(t) ofC 1 is ≥ ct d−1 d for 0 < t < |C 1 |/2, which is a known fact (see [10]).
Let us finish the demonstration of Claim 4.11. We now have to show that the sets C n cannot be-up to a set of bounded measure-inverse images of balls by some large-scale equivalence. So let (X ′ , d, µ) be a (M)-space and let f : X → X ′ be a large-scale equivalence.
Let us consider two points of C n of respectively maximum and minimum x 2 . The distance of each of these points to C c is ≥ n/2 and yet, every 1-chain joining them must pass through C n ∩ {x 2 = 0} whose points are at distance ≤ 2 log n from C c . But this is impossible for a ball 4.4. The instability of (IB) under quasi-isometry between graphs of polynomial growth. Theorem 4.13. Let d be an integer ≥ 2. There exists two graphs X and X ′ quasi-isometric, of polynomial growth of degree d and with bounded or unbounded profile, such that X satisfies (IB) but not X ′ .
Like in the examples of the two previous sections, we will build a graph X removing some edges from Z d : for n ∈ N, let A n be the ball of radius n whose center belongs to the axis Z.e 1 in such a chain that A n+1 is at distance 2 n from A n . We then remove all the edges of the boundary of A n but those belonging to the line Z.e 1 . We write A for the union of A n . The graph X ′ is obtained from X by taking its image by the linear map fixing the first coordinate and acting on the orthogonal as an homothetic transformation of ratio 4 (it is clear that it is a quasi-isometry). More precisely, we replace each edge of X parallel to the first axis, by a chain of length 2 also parallel to the first axis. Write A ′ for the image of A.
Remark 4.14. In the previous example, the profile is bounded. Nevertheless, one can slightly modify the construction in order to get an unbounded profile: for instance, removing only edges of the boundary of A n at distance ≥ log n from the axis Z.e 1 (instead of those which are outside of this axis).
Claim 4.15. The graphs X and X ′ have polynomial growth of degree d.
As these graphs are subgraphs of Z d , their volume growths are less than the one of Z d . The converse inequality will follow from the fact that in X ′ , the profile restricted to balls is strong and from the fact that X and X ′ are quasi-isometric. Proof : It is clear by construction that the A n are balls and that their boundaries have bounded volume.
Claim 4.17. In X ′ , the profile restricted to balls is strong: Proof : Remark that Lemma 4.2 stays true in this context. Let B = B(x, r) be a ball of the graph X ′ . We have to show that there exists a constant c > 0 such that According to Lemma 4.2, we can assume that B ⊂ A. Thus, there exists n 0 such that B ⊂ A n0 . Let us embed Z d into R d . Let us replace the discrete polyhedron A n and B by their convex hullsÃ n andB in R d . LetX be the space obtained removing from R d (Euclidian) the points of the Euclidian boundary ofÃ n (for all n) but the two ones belonging to the axis R.e 1 (resp. those at distance ≤ log n of the axe) for the case of bounded profile (resp. for the case of unbounded profile). Let us equipX -seen as a subset of R d -with Lebesgue measure and with the geodesic metric d(x, y) = inf γ l(γ) with γ taking values in the set of arcs joining x to y inX, l(γ) being the Euclidian length of γ.
The embedding j of X intoX we obtain like this is clearly a large-scale equivalence. For simplicity, we will write |A| for the (Lebesgue) measure of a subset A ofX. On the other hand, note that ∂ 10B contains [j(B(x, r))] 1 \ [j(B(x, r − 2))] 1 , which by Proposition 2.6 has same measure (up to multiplicative constant) as ∂B. The same holds forB and B. Moreover, sincẽ B and A n0 are convex polyhedra, it is clear that the 10-boundary ofB has same measure (up to multiplicative constants) as its Euclidian boundary (whose measure is the limit when h → 0 of |∂ hB |/h). Write: Consequently, it is enough to show that there exists c > 0 such that Note that by homogeneity, the quantity Q = 1 r d−1 |∂ euclB | only depends on the ratio n/r. Fix n = n 0 . For r small enough (let us say ≤ r c for some r c > 0), B never meets two parallel faces: Q stays larger than a constant > 0 (i.e. profile of a 1/2 d−1 'th of space of R d ). By compactness, it follows that Q reaches its minimum when x and r vary under the conditions: r c ≤ r ≤ n 0 /2. On the other hand, asB is strictly included inÃ n0 , this minimum has to be > 0. The ratio Q is therefore larger than a constant c ′ > 0. finally, there is a constant c > 0 such that: So we are done.

Asymptotic isoperimetry of connected subsets.
Recall that we say that a subset A of a metric space is connected if there does not exist a nontrivial partition Let (X, d, µ) be a metric measure space. Write C for the set of connected subsets of finite measure of X.
(i) Let X be such that the measure of balls of radius 1/2 are minorated by a > 0. Suppose that I(t) = o(t). Then there exists a positive and increasing sequence (t i ) tending to infinity such that I ↓ C (t i ) = I(t i ).
(ii) Assume that X is a doubling (M)-space and has a strong profile. Then I c ≈ I.
(iii) Let d be an integer ≥ 2. there exists a graph X of polynomial growth of degree d and a increasing sequence of integers (N n ) such that I ↓ C (N n ) = o(I(N n )). Proof : Note that (ii) follows from Corollary 3.5 and from the fact that property (M) implies that balls are connected.
Let us show the first assertion of the theorem. Suppose that there exists T > 0 such that ∀t ≥ T , I(t) < I ↓ C (t). We will show that it implies that: Write t m for the upper bound of the set of t such that ∀s ≤ t, one has: I(s) ≥ a s T . Since I is nondecreasing, if t m is finite, then it is a maximum.
Remark that t m ≥ T since the boundary of every nonempty subset of X contains a ball of radius 1/2 (see Proposition 2.6) and therefore has measure ≥ a.
Suppose by absurd that t m is finite. By definition of t m , for all s > t m there exists a subset A such that: µ(A) ≥ s and: µ(∂A) < as/T. Moreover, since t m ≥ T , we can suppose that: µ(∂A) < I ↓ C (s) (in particular, A is not connected).
It follows that there exists a smallest positive integer k such that there exist t m ≤ s ≤ t m + T /2 and a subset A of measure ≥ s, with k connected components and whose boundary has measure < min{I ↓ C (s), sa/T }. Let A be such a subset. Note that k ≥ 2. Thus, we have: Since k is minimal, one has, for i = 1, 2: Indeed, if for instance, one had µ(A 1 ) ≥ t m , then since the boundary of A 2 has measure ≥ a, one would have: µ(∂A 1 ) ≤ (t m + T /2) a T − a = t m a T − a/2 < t m a T .
Therefore, as I ↓ C (t m ) ≥ I(t m ) ≥ t m a/T , one would also have: µ(∂A 1 ) < I ↓ C (t m ). But then, by minimality of k, A 1 should have at least k connected components, which is absurd since it has strictly less components than A.
But, by definition of t m , this implies that: In order to show the second assertion of the theorem, we proceed as in the previous sections: we start from the graph Z d , and then we remove some edges. Let us consider the following family of cubes (C m n ) 0≤m≤n−1,n∈N * of Z d : the C m n are Euclidian cubes of edges' length 2 2 n whose centers are disposed along the axis Z.e 1 as follows: C m+1 n is the image of C m n by the translation of vector n2 2 n .e 1 and C n−1 n and C 1 n+1 are at distance (n + 1)2 2 (n+1) to one another. To build the graph X, we remove all the edges joining C m n to the rest of the graph but those which have a vertex belonging to the Euclidian cube c m n of dimension d − 1 of the boundary of C m n , of volume 2 n 2 and centered in one of the two intersection points of C m n with the axis Z.e 1 . Write C for the union of cubes C m n . Claim 5.2. The growth in X is polynomial of degree d.
Proof : Let B = B(x, r) be a ball. Let us prove that |B| ≥ 2 −100d r d . If the center of B doesn't belong to any C m n , it is clear. Suppose therefore that x ∈ C m0 n0 for integers n 0 and m 0 < n 0 . Write D n0 for the diameter of C m0 n0 . If r ≥ 3D n0 , then B contains B(y, r/2) with y belonging to no C m n . So we are brought back to the previous case. In the other case, the conclusion follows from the following trivial fact: in Z d , if r ≤ n, the volume of the intersection of a cube C of edges' length equal to n with a ball of radius r ≤ n and of center x ∈ C is ≥ 2 −d |B(x, r)| ≥ 2 −10d r d . Indeed, the worst case is when x is a corner of the cube. Claim 5.3. Take N n = n2 2 n . Then I(N n ) = o(I ↓ C (N n )). Proof : Let us consider the set C n = ∪ m C m n . Its volume is equal to N n and its boundary has volume equal n2 n 2 . On the other hand, let n 1 be an integer and let P be a connected subset of volume ≥ N n1 . We want to show that |∂P | ≥ c2 (n1+1) 2 , for a constant c > 0, which is clearly enough to conclude.
Thanks to the following lemma, the only remaining case to consider is when P meets a cube C m n . But, because of the large distance between two such cubes, we can assume that P meets only one of these cubes, say C m0 n0 . If |P ∩C c | ≥ |P |/2, then the lemma applied to P ∩C c allows to conclude. Suppose therefore that |P ∩ C| ≥ |P |/2. This implies in particular that n 0 ≥ n 1 + 1. We then remark that |∂(P ∩ C m0 n0 )| ≤ |∂P |. Indeed, let π be the orthogonal projection onto the hyperplane containing c m0 n0 , then every point of c m0 n0 ∩ P admits un antecedent by π belonging to the boundary of P . So we can assume that P ⊂ C m0 n0 . If |P | ≤ 3/2|C m0 n0 |, then there exists c > 0 such that: (isoperimetry in the full Euclidian cube: see [10]). Otherwise, assume that |P | ≥ 3/2|C m0 n0 | and write Q = C m0 n0 \ P . • If the volume of Q is ≥ D n0 /2 where D n0 is the diameter of C m0 n0 , then 5.2 applied to Q implies that: But, the boundary of Q is-up to points belonging to c m0 n0 (whose cardinal is negligible compared to c2 2 n 0 −1 )-equal to the boundary volume of P . So we are done.