Continuum Percolation for Quermass Model

The continuum percolation for Markov (or Gibbs) germ-grain models is investigated. The grains are assumed circular with random radii on a compact support. The morphological interaction is the so-called Quermass interaction defined by a linear combination of the classical Minkowski functionals (area, perimeter and Euler-Poincar\'e characteristic). We show that the percolation occurs for any coefficient of this linear combination and for a large enough activity parameter. An application to the phase transition of the multi-type Quermass model is given.


Introduction
The germ-grain model is built by unifying random convex setsthe grains -centered at the pointsthe germs -of a spatial point process. It is used for modelling random surfaces and interfaces, geometrical structures growing from germs, etc. For such models, the continuum percolation refers mainly to the existence of an unbounded connected component. This phenomenon expresses some macroscopic properties of materials as permeability, conductivity, etc. Moreover, it turns out to be an efficient tool to exhibit phase transition in Statistical Mechanics [2,4]. For theses reasons, the continuum percolation has been abundantly studied since the eighties and the pioneer paper of Hall [8].
When the grains are independent and identically distributed, and the germs are given by the locations of a Poisson point process (PPP), the germ-grain model is known as the Boolean model. In this context, the continuum percolation is well understood; see the book of Meester and Roy [13] for a very complete reference. One of the first results is the existence of a percolation threshold z * for the intensity parameter z of the stationary PPP: provided the mean volume of the grain is finite, percolation occurs for z > z * and not for z < z * .
Because of the independence properties of the PPP, the Boolean model is sometimes caricatural for the applications in Biology or Physics. Mecke and its coauthors [11,12] have mentioned the need of developing, via Markov or Gibbs process, an interacting germ-grain model in which the interaction would locally depend on the geometry of the set. For this purpose, let us cite the Widom-Rowlinson model [16], the area interaction process [1] and the morphological model [12]. Thus, Kendall, Van Lieshout and Baddeley suggested in [9] a generalization of the previous models, called the Quermass Interaction Process. In this model, the formal Hamiltonian is a linear combination of the fundamental Minkowski functionals, namely in R 2 the area A, the perimeter L and the Euler-Poincaré characteristic χ: H = θ 1 A + θ 2 L + θ 3 χ .
The existence of infinite volume Gibbs point processes for the Hamiltonian H has been recently proved in [3]. This paper focuses on the continuum percolation for such processes.
The existence of a percolation threshold z * for the Boolean model relies on a basic (but essential) monotonicity argument: see [13], Chapter 2.2. This argument fails in the case of Gibbs point processes with Hamiltonian H. So, no percolation threshold can be expected in our context. However, other stochastic arguments as stochastic domination or FKG lead to percolation results. In [2], Chayes et al prove that percolation occurs for z large enough and θ 2 = θ 3 = 0. To our knowledge, the percolation phenomenon for other values of parameters θ 1 , θ 2 , θ 3 has not been investigated yet.
Our main result (Theorem 1) states that, for any θ 1 , θ 2 , θ 3 (positive or negative), percolation occurs with probability 1 for z large enough. The only assumption bears on the random radii of the circular grains: they have to belong to a compact set not containing 0. The proof of this theorem is relatively easy in the case θ 3 = 0. Indeed, the local energy h((x, R), ω)-the energy variation when the grainB(x, R) is added to the configuration ωis uniformly bounded (see Lemma 4.12) and by classical stochastic comparison with respect to the Poisson Process the result follows. So the main challenge of the present paper is the proof of Theorem 1 when θ 3 = 0. In this setting, the local energy becomes unbounded from above and below and classical stochastic comparison arguments for point processes fail. In interpreting the percolation in our model via a site percolation model (see the beginning of Section 4), we prove the result thanks to a stochastic domination result for graphs due to Liggett et al (Theorem 1.3 in [10] or Lemma 4.2 below). An arduous control of the hole number variation, when a new grain is added, is the main technical issue. We prove essentially that this variation is moderate for a large enough set of admissible locations of grains. Let us mention that our proof is inspired by the one of Proposition 3.1 in [4].
Following [2,4], we use our percolation result (Theorem 1) to exhibit a phase transition phenomenon for Quermass model with several type of particles (Theorem 2).
Our paper is organized as follows. In Section 2, the Quermass model and the main notations are introduced. The local energy h((x, R), ω) is defined in (2.3). Section 3 contains the results of the paper. Section 3.2 is devoted to the case θ 3 = 0 and Section 3.3 to the phase transition result. The proof of Theorem 1 is developed in Section 4.

Notations
We denote by B(R 2 ) the set of bounded Borel sets in R 2 with a positive Lebesgue measure. For any Λ and ∆ in B(R 2 ), Λ ⊕ ∆ stands for the Minkoswki sum of these sets. Let 0 ≤ R 0 ≤ R 1 be some positive reals and E be the product space A configuration ω is a subset of E which is locally finite with respect to its first coordinate: #(ω ∩ E Λ ) is finite for any Λ in B(R 2 ). The configuration set Ω is endowed with the σ-algebra F generated by the functions ω → #(ω ∩ A) for any A in σ(E). We will merely denote by ω Λ instead of ω ∩ E Λ the restriction of the configuration ω (with respect to its first coordinate) to Λ. Moreover, for any (x, R) in E, we will write ω ∪ (x, R) instead of ω ∪ {(x, R)}.
A configuration ω ∈ Ω can be interpreted as a marked configuration on R 2 with marks in [R 0 , R 1 ]. To each (x, R) ∈ ω is associated the closed ballB(x, R) (the grain) centered at x (the germ) with radius R. The germ-grain surfaceω is defined as

Quermass interaction
Let us define the Quermass interaction as in Kendall et al. [9]. The energy (or Hamiltonian) of a finite configuration ω in Ω is defined by can be decomposed as in (2.1). This universal representation justifies the choice of the Quermass interaction for modelling mesoscopic random surfaces [11,12]. The energy inside Λ ∈ B(R 2 ) of any given configuration ω in Ω (finite or not) is defined by where ∆ is any subset of R 2 containing Λ ⊕ B(0, 2R 1 ). By additivity of functionals A, L and χ, the difference H Λ (ω) does not depend on the chosen set ∆. Let us end with defining the local energy h((x, R), ω) of the marked point (x, R) ∈ E (or of the associated ballB(x, R)) with respect to the configuration ω: for any Λ ∈ B(R 2 ) containing x. Remark this definition does not depend on the choice of the set Λ. The local energy h((x, R), ω) represents the energy variation when the ball B(x, R) is added to the configuration ω.

The Gibbs property
Let Q be a reference probability measure on [R 0 , R 1 ]. Without loss of generality, R 0 and R 1 can be chosen such that, for every ε > 0, Let z > 0. Let us denote by λ the Lebesgue measure on R 2 and by π z the PPP on E with intensity measure zλ ⊗ Q. Under π z , the law of the random surfaceω is the stationary boolean model with intensity z > 0 and distribution of radius Q. Finally, for any Λ ∈ B(R 2 ), let us denote by π z Λ the PPP on E Λ with intensity measure zλ Λ ⊗ Q, where λ Λ is the restriction of the Lebesgue measure λ to Λ. Definition 2.1. A probability measure P on Ω is a Quermass Process for the intensity z > 0 and the parameters θ 1 , θ 2 , θ 3 if P is stationary and if for every Λ in B(R 2 ), for every bounded positive measurable function f from Ω to R, The equations (2.5)-for all Λ ∈ B(R 2 ) -are called DLR for Dobrushin, Landford and Ruelle. They are equivalent to: for any Λ ∈ B(R 2 ), the law of ω Λ under P given ω Λ c is absolutely continuous with respect to the Poisson Process π z Λ with the local density See [15] for a general presentation of Gibbs measures and DLR equations. The existence, the uniqueness or non-uniqueness (phase transition) of Quermass processes are difficult problems in statistical mechanics. The existence has been proved recently in [3], Theorem 2.1 for any parameters z > 0 and θ 1 , θ 2 , θ 3 in R . A phase transition result is proved in [2,6,16] for R 0 = R 1 , θ 2 = θ 3 = 0 and for θ 1 = z large enough.

Percolation occurs
We say that percolation occurs for a given configuration ω ∈ Ω if the subsetω of R 2 contains at least one unbounded connected component. The set of configurations such that percolation occurs is a translation invariant event. Its probability, called the percolation probability, equals to 0 or 1 for any ergodic Quermass process. However, the Quermass processes are not necessarily ergodic (they are only stationary) and their percolation probabilities may be different from 0 and 1. Besides, in [2], Chayes et al have built two Quermass processes, both corresponding to θ 2 = θ 3 = 0 and θ 1 = z large enough, whose percolation probabilities respectively equal to 0 and 1. Since any mixture of these two processes is still a Quermass process, the authors obtain Quermass processes whose percolation probabilities equal to any value between 0 and 1. Our main result states that percolation occurs with probability 1 for any (ergodic or not) Quermass process whenever the intensity z is large enough. Theorem 1. Let R 0 > 0 and θ 1 , θ 2 , θ 3 ∈ R. There exists z * > 0 such that for any Quermass process P associated to the parameters θ 1 , θ 2 , θ 3 and z > z * , percolation occurs P -almost surely.
The proof of Theorem 1 is based on a discretization argument which allows to reduce the percolation problem from the (continuum) Quermass model to a site percolation model on the lattice Z 2 (up to a scale factor). This proof is rather long and technical so it is addressed in Section 4. Let us point out here that our theorem does not claim z * is a percolation threshold. In other words, for z < z * , the percolation may be lost and recovered on different successive ranges. Another natural question involves the number of unbounded connected components. Following the classical arguments for continuum percolation, we prove that this number is almost surely equal to zero or one. Proof. It is well-known that any Gibbs measure is a mixture of extremal ergodic Gibbs measures. For each ergodic Quermass process P , the number of connected component is almost surely a constant in N ∪ {+∞}. For any Λ ∈ B(R 2 ), thanks to the DLR equations (2.5), it is easy to prove that the law of ω Λ under P is equivalent to π z Λ . Therefore, in following the general scheme of the proof of Theorem 2.1 in [13], we show that the number of connected components is necessary 0 or 1.

Percolation when θ 3 = 0
In the particular case θ 3 = 0, Theorem 1 can be completed and proved in a simple way.
First, let us recall the definitions involving the stochastic domination for point processes. We follow the notations given in [5]. An event A in F is called increasing if for every ω ∈ A and any ω ′ ∈ Ω containing ω then ω ′ ∈ A too. Let P and P ′ be two probability measures on Ω. We say that P is dominated by P ′ , denoted by P P ′ , if for every increasing event A ∈ F, P (A) ≤ P ′ (A). In this section, we focus our attention on the increasing event "there exists an unbounded connected component".
Let P be any Quermass process and assume θ 3 = 0 and R 0 > 0. Thanks to Lemma 4.12, the local energy can be uniformly bound: there exist constants C 0 and C 1 such that for any (x, R) ∈ E and ω ∈ Ω, Combining (3.1) and Theorem 1.1 in [5], we get the following stochastic dominations: Now, the (stationary) Boolean models corresponding to π ze −C 1 and π ze −C 0 admit positive and finite percolation thresholds (see [14], Chapter 3). It follows : there exist constants z 0 , z 1 such that for any Quermas Process P associated to parameters z, θ 1 , θ 2 and θ 3 = 0, the percolation occurs P -almost surely if z > z 1 and does not occur P -almost surely if z < z 0 .
Proposition 3.2 improves Theorem 1 in the case θ 3 = 0 since it ensures the existence of a subcritical regime.
It is worth pointing out here that the uniform bounds in (3.1) do not hold whenever θ 3 = 0. Precisely, this is the hole number variation which cannot be uniformly bounded.

Phase transition for multi-type Quermass Process
In this section, the multi-type Quermass model is introduced and a phase transition is exhibited, i.e. the existence of several Gibbs processes for the same parameters is proved.
Let K be a positive integer. The K-type Quermass model is defined on the space Each disc is now marked by a number specifying its type. We don't give the natural extension of the notations involving the sigma-field and so on. The following Quermass energy function is defined such that all discs of a connected component have the same number. This is a non-overlapping multi-type germ-grain model. Precisely the energy of a finite configuration ω is now given by where φ is an hardcore potential equals to infinity on ] − ∞, 0] and zero on ]0, +∞]. The energy inside Λ ∈ B(R 2 ) of any finite or infinite configuration ω is defined as in (2.2) with the convention +∞ − ∞ = +∞. The definition of the K-type Quermass process via the DLR equations follows as in Definition 2.1. The proof of the existence of such processes is similar to the one of the existence of Quermass process. See Theorem 2.1 of [3] for more details. Here is our phase transition result: For any θ 1 , θ 2 and θ 3 in R, there exists z 0 > 0 such that, for any z > z 0 , there exist several K-type Quermass Processes. The phase transition occurs.
The proof essentially follows the scheme of the one of Theorem 2.2 of [2] or Theorem 1.1 of [4]. It is based on a random-cluster representation (or Gray Representation) analogous to the Fortuin-Kasteleyn representation of the Potts model. The existence of an unbounded connected component allows to prove the existence of a K-type Quermass process in which the density of particles of a given type is larger than the ones of the other types. By symmetry of the types, we prove the existence of at least K different K-type Quermass processes.
4 Proof of Theorem 1
For any x ∈ (6ℓ Z) 2 , let τ x be the translation operator on the configuration set E defined by (y, R) ∈ τ x ω if and only if (y + x, R) ∈ ω. Hence, we can define the translated indicator function ξ x of ξ on the translated box ∆ x = x + ∆ by ξ x (ω) = ξ(τ x ω). Let us remark that ξ x (ω) only depends on the restriction of the configuration ω to the box ∆ x . Moreover, thanks to the stationary character of the Quermass process P , the random variables ξ x , x ∈ (6ℓ Z) 2 , are identically distributed. They are dependent too. Let us consider x, y ∈ (6ℓ Z) 2 such that y = (6ℓ, 0) + x. The boxes ∆ x and ∆ y have in common a cardinal box, i.e. x + B E = y + B W . So, the condition ξ x (ω) = ξ y (ω) = 1 ensures that the cardinal boxes of ∆ x and ∆ y are connected together through the restriction ofω to ∆ x ∪ ∆ y . The same is true when y = (0, 6ℓ) + x. This induces a graph structure on the vertex set V = (6ℓ Z) 2 : for any x, y ∈ V , {x, y} belongs to the edge set E if and only if The graph (V, E) is merely the square lattice Z 2 with the scale factor 6ℓ. The family {ξ x , x ∈ V } provides a site percolation process on the graph (V, E). It has been built so as to satisfy the following statement.   Whenever the Quermass process P satisfies (4.1) for that p, then combining Lemmas 4.1 and 4.2 percolation occurs P -a.s. Therefore it remains to show that for any p > 0, hypothesis (4.1) holds for z large enough.
The next result claims that each Borel set of R 2 , sufficiently thick in some sense, contains at least one element of the configuration ω with a probability tending to 1 as the intensity z tends to infinity. It will be proved at the end of this section. Lemma 4.3. Let V ⊂ R 2 such that there exist U ∈ B(R 2 ) with positive Lebesgue measure and ε > 0 satisfying U ⊕B(0, R 1 + R 0 + ε) ⊂ V . Then there exists a constant C > 0, depending on λ(U ) and ε, such that for any configuration ω ∈ Ω and for any z > 0, Since the Quermass process P is stationary, it is sufficient to prove (4.1) with x = (0, 0). So, we focus our attention on the diamond box ∆ = ∆ (0,0) and use Lemma 4.3 to check that condition (C1) is fulfilled in this box. Since B N , B S , B E and B W are sufficiently thick (with side length ℓ > 2R 1 + 2R 0 ), it follows for any i ∈ {N, S, E, W}. So the conditional probability that ω satisfies (C1) is larger than 1 − 4Cz −1 . The equation N ∆ cc (ω) = 0 forces the box B N (for instance) to be empty of points of the configuration ω. Hence, Checking that condition (C2) is fulfilled in the diamond box ∆ needs what we call the Connection Lemma (Lemma 4.4). This result states the conditional probability that N ∆ cc (ω) is larger than 2 converges to 0 uniformly on the configuration outside ∆. This is the heart of the proof of Theorem 1. Its technical proof is given in Section 4.2.
Lemma 4.4 (The Connection Lemma). There exists a constant C ′ > 0 such that for any configuration ω ∈ Ω and for any z > 0, The above inequalities and the Connection Lemma imply that conditions (C1) and (C2) are fulfilled in ∆ with a probability tending to 1 as z tends to ∞: The hypothesis (4.1) then follows. Let x be a vertex of the graph (V, E) which is not a neighbor of (0, 0). By construction, the box ∆ x is included in ∆ c = ∆ c (0,0) (since ∆ is an open set). This means the random variable ξ x is measurable with respect to the σ-algebra induced by the configurations restricted to ∆ c (0,0) . So, and the hypothesis (4.1) holds with x = (0, 0) and any p ∈ [0, 1[, provided the intensity z is large enough. This ends the proof of Theorem 1.

Outline
Let us recall that N ∆ cc (ω) denotes the number of connected components ofω ∆ having at least one ball centered in one of the four cardinal boxes B N , B S , B E or B W . In this section, we assume N ∆ cc (ω) ≥ 2. Our strategy consists in exhibiting a subset B of the diamond box ∆ in which ω B = ∅. Moreover, for x ∈ B, if we are able to control uniformly the energy H B ((x, R) ∪ ω B c ) on ω B c , then the configuration ω should contain a point centered in B with large probability as z tends to infinity. This leads to the Connection Lemma.
For x ∈ B, let us denote by N hol ((x, R), ω B c ) the hole number variation when the ball B(x, R) is added to the configuration ω B c . This quantity is central in our proof. Indeed, a first upperbound for the energy H B ((x, R) ∪ ω B c ) is given by Lemma 4.12: where K is a positive constant only depending on parameters θ 1 , θ 2 , θ 3 and radii R 0 , R 1 . So, to upperbound the energy H B ((x, R) ∪ ω B c ) it suffices to upperbound the number of created holes (resp. deleted holes) when θ 3 is negative (resp. positive). This is the reason why the proof of the Connection Lemma differs according to the sign of the parameter θ 3 .

When θ 3 is negative
Let ω be a configuration and α be a positive real number. A couple (x, R) ∈ R 2 × [R 0 , R 1 ] is said good if all the connected components of the setω ∆ ∩B(x, R) have an area larger than α. These couples are well-named because adding a ballB(x, R) to the configuration ω ∆ , with a good couple (x, R), does not create too many holes.
Proof. The number of created holes when the ballB(x, R) is added to ω ∆ is smaller than the number of connected components of the setω ∆ ∩B(x, R). Since (x, R) is good, all these connected components have an area larger than α. So, there are at most πR 2 /α such connected components.
Let us denote by Bad(ω ∆ , α) the following set: Lemma 4.6. The area of the set Bad(ω ∆ , α) tends to 0 as α and ε tend to 0, uniformly on the configuration ω ∆ .
Lemma 4.6 will be proved at the end of this section. Thanks to Lemmas 4.5 and 4.6, we are now able to prove the Connection Lemma. First, we need a family of (small) nonoverlapping squared boxes whose union covers the convex hull of the boxes B N , B S , B E and B W . Precisely, for κ > 0, let us consider a subset B of {v The family B is made up of at most c κ = κ −2 A(∆) elements. The hypothesis N ∆ cc (ω) ≥ 2 ensures the existence of two elements (x 1 , ·) and (x 2 , ·) of ω, whose centers x 1 and x 2 are in the union of the four cardinal boxes B N , B S , B E and B W , and whose ballsB(x 1 , ·) andB(x 2 , ·) belong to two different connected components ofω, say respectively C 1 and C 2 . Let [x 1 , x 2 ] be the segment in R 2 linking x 1 with x 2 and d be the euclidean distance on R 2 . The continuous application and d(x,ω \ C 1 ) are equal (and positive). Hence, the ballB(x, R 0 ) does not contain any point of ω ∆ . Moreover, since x is in the convex hull of the boxes B N , B S , B E and B W , then it belongs to one box of the family B, say B. With κ < R 0 / √ 2, the box B is contained in B(x, R 0 ). Consequently, ω B is empty : For a given box B ∈ B, let us consider the (random) set U (ω ∆\B ) of points x ∈ B such that, for any radius R ∈ [R 0 , R 0 + ε], the couple (x, R) is good: .
On the one hand, using (4.5), θ 3 ≤ 0 and Lemma 4.5, we get where M = M (R 1 , α) denotes the upperbound given by Lemma 4.5. On the other hand, Lemma 4.6 implies the area of U (ω ∆\B ) is larger than κ 2 /2 for α and ε small enough, uniformly on the configuration ω ∆\B . It follows: In the previous inequality, replacing e −zκ 2 Z B (ω B c ) −1 with the conditional probability P (ω B = ∅|ω B c ), we obtain .
Finally, the Connection Lemma derives from the above upperbound, (4.6) and with .
Then there exists a positive function g(ε, α) tending to 0 as α and ε tend to 0, such that | d(x,B(y, ·)) − R 0 | ≤ g(ε, α) . (4.8) The function g is also allowed to depend on radii R 0 and R 1 . The topological boundary ∂ω ∆ is composed of a finite number of arcs. Let a be one of them. This arc is generated by an element of the configuration ω ∆ , say (y, ·). Now, we can define the circular strip S g (a) of width 2g(ε, α) by Let (y, ·) be the element of the configuration ω ∆ generating the arc a. Let S(a) be the semi-infinite cone centered at y and with arc a (i.e. the union of semi-line [y, y ′ ) for y ′ ∈ a). Then, x necessarily belongs to S(a). Indeed, the opposite situation could lead to the existence of another arc a ′ satisfying d(x, a ′ ) < d(x, a). To sum up, x is in the semi-infinite cone S(a) and the area ofB(x, R) ∩B(y, ·) is positive and smaller than α. So x satisfies (4.8) and then belongs to S g (a).
Proof of Lemma 4.6. Let a be an arc of the boundary ∂ω ∆ . Some geometrical considerations allow to bound the area of the circular strip S g (a): where length(a) denotes the length of the arc a. We deduce from this bound and Lemmas 4.7 and 4.11: This latter upperbound does not depend on the configuration ω ∆ . So, this ends the proof of Lemma 4.6.

When θ 3 is positive
In this section, we still assume that N ∆ cc (ω) is larger than 2. But this time, our aim consists in upperbounding the number of deleted holes when the ballB(x, R), x ∈ B, is added to the configuration ω B c . The existence of a suitable set B derives from Lemma 4.8. Its proof is rather long and technical, mainly because of the uniformity of ρ > 0 with respect to the configuration ω.

not (totally) contain any hole ofω.
Let us first explain how the Connection Lemma straight derives from Lemma 4.8. As in Section 4.2.2, we need the family B of non-overlapping squared boxes of length side κ. But here, B is required to cover a little bit more, i.e.
and parameters κ and ε are chosen small enough so that √ 2κ + ε < ρR 0 (4.11) (where ρ is given by Lemma 4.8). Thanks to statement (i) and (4.10), the point O belongs to a box B ∈ B. Thanks to (ii), (iii) and (4.11), ω B is empty andω B c has no hole in B := B ⊕ B(0, R 0 + ε). Hence, Let us pick a box B ∈ B, a couple (x, R) ∈ B × [R 0 , R 0 + ε] and assume thatω B c has no hole in B. Then, no hole is deleted whenB(x, R) is added to ω B c . So, the hole number variation N hol ((x, R), ω B c ) is nonnegative. Combining with θ 3 ≥ 0 and (4.5), the energy H B ((x, R) ∪ ω B c ) is smaller than K and we finish the proof of the Connection Lemma as in Section 4.2.2. First, Thus, replacing e −zκ 2 Z B (ω B c ) −1 by the conditional probability P (ω B = ∅|ω B c ), we get Finally, the Connection Lemma derives from the above upperbound, (4.12) and with where c κ still denotes the number of boxes contained in the family B.
where C 1 denotes a connected component ofω ∆ counting by N ∆ cc (ω). Two cases will be considered in the following. In the first oned ≥ 1 2 R 0 -the connected components ofω ∆ are far away from O ′ . So do their holes. Then, the choice O = O ′ is appropriate. In the second cased ≤ 1 2 R 0 -we exhibit a region close to O ′ without hole and choose a suitable point O inside. About the radius ρ, it will be proved in the sequel that any positive real number such that (1 + ρ) 2 < 1 + 1 4 , (4.14) is suitable. For instance, ρ = 0.01 satisfies these three conditions.
This means that the hole T is outside the ball B(O ′ , (1 + ρ)R 0 ). Now, assume that O ′ is in T . Since O ′ is equidistant from two connected components ofω ∆ then one of them is inside the hole T . Hence, T is too large to be totally covered by the ball B(O ′ , (1 + ρ)R 0 ). Consequently, O ′ also satisfies (iii).
In the same way, let us consider a point y 2 such thatB(y 2 , R 0 ) is included inω ∆ \ C 1 and The region without hole, mentioned at the beginning of the current section and which we need, is built from points y 1 and y 2 . See  Proof of Lemma 4.9. The closest hole T to the segment [y 1 , y 2 ] is obtained by pressing a ball with radius R 0 againstB(y 1 , R 0 ) andB(y 2 , R 0 ). If l denotes the distance between T and [y 1 , y 2 ] then 2l is the distance between the center of this pressing ball and [y 1 , y 2 ]. Pythagoras Theorem gives (2l In the worst case, h = 1 2 R 0 . Hence, l is always larger than √ 7 4 R 0 , which is the distance between D and D ′ . To complete the proof, let us add there is no hole in the ballsB(y 1 , R 0 ) andB(y 2 , R 0 ) since they are totally covered byω ∆ .
The idea to conclude the proof can be sum up as follows. The region H is sufficiently thick to contain strictly more than half of a ball with radius (1 + ρ)R 0 . Hence, the part of this ball outside H (this is the hatched region on Figure 2) has a diameter smaller than By convexity and statement (a), any point of the segment [z 1 , z 2 ] is at distance from O ′ larger than Combining statement (c) with i = 1 and Lemma 4.14, we check there is no hole ofω ∆ \ C 1 in the ball B(z 1 , (1 + ρ)R 0 ). Let us run the center of a ball with radius (1 + ρ)R 0 along the segment [z 1 , z 2 ] from z 1 to z 2 until that ball meets a hole ofω ∆ \ C 1 . Two cases can be distinguished.
• This meet does not happen. Then, the ball B(z 2 , (1 + ρ)R 0 ) does not contain any hole ofω ∆ \ C 1 . It does not contain any hole of C 1 either thanks to statement (c) with i = 2 and Lemma 4.14. In this case, O = z 2 satisfies the property (iii) of Lemma 4.8.
• This meet happens: let O be the corresponding center and T be the corresponding hole ofω ∆ \ C 1 . As just before, the ball B(O, (1 + ρ)R 0 ) does not still contain any hole ofω ∆ \ C 1 . Denote by C the part of this ball outside H: On the one hand, the diameter of C is smaller than 2R 0 thanks to (4.15). On the other hand, C is pressed against the hole T (there is no hole in H); see i.e. statement (a). Second, the ballsB(z i , ρR 0 ) which are included in V, are also included inB(O ′ , R 0 + d). So does the set [z 1 , z 2 ]⊕ B(0, ρR 0 ) by convexity. Statement (b) is proved. It remains to prove statement (c). Let us introduce the orthogonal projection h 1 of z 1 over the infinite line D (see Figure 2). Using d(M, Thanks to (4.16), statement (c) follows.

Proofs of geometrical lemmas
Lemma 4.11. Let ∆ be a bounded closed convex set. For any configuration ω, let us denote by L ∆ (ω) the perimeter ofω viewed through ∆: where length(∂∆ ∩ω) denotes the lentgh of the boundary of ∆ which is inside the setω. Then, Proof. The boundary ofω viewed through ∆ corresponds to a finite union of arcs, say (a i ) 1≤i≤n . For each arc a i , coming from the ball B(x i , R i ), we consider the circular strip S(a i ) of width R 0 defined by Let us notice that the sets (S(a i )) 1≤i≤n are disjoint. Indeed, let suppose that there exists x ∈ S(a i ) ∩ S(a j ) for some i = j. Without restriction, we can assume that the distance between x and a i is smaller than or equal to the distance between x and a j . Let y be the point on a i such that this distance is equal to |y − x|. Then, y has to be strictly included in the ball B(x j , R j ) which contradicts the fact that y is on the boundary ofω. This allows to compare the sum of the areas of (S(a i )) 1≤i≤n with A(ω): In the same way, we consider the perimeter variation L((x, R), ω) and the connected component number variation N cc ((x, R), ω). The following inequalities hold.
(4.17) R), ω) ≤ 1 . Proof. Inequalities (4.17), upperbounds of (4.18) and (4.19) are obvious. The border length ofω which is lost when the ballB(x, R) is adding can be interpreted as the perimeter of ω viewed throughB(x, R) , i.e. as LB (x,R) (ω). Thanks to Lemma 4.11, it is smaller than This gives the lowerbound of (4.18). It remains to lowerbound N cc ((x, R), ω). For that purpose, the number of deleted connected components whenB(x, R) is adding toω, is smaller than the number of non-overlapping balls which overlapB(x, R). This number is at most 2π(R 1 + R 0 ) 2R 0 . Necessarily, y is on the boundary of two balls B(z, R) and B(z ′ , R ′ ) of C. Since x belongs neither to C nor to T , at least one of z − y or z ′ − y has a nonnegative scalar product with x − y. Say z − y. Given |x − z| and |y − z|, the distance |x − y| is minimal when the vectors z − y and x − y are orthogonal. Hence, using |x − z| ≥ d(x, C) + R 0 and |y − z| ≥ R 0 , it follows from Pythagoras Theorem that which concludes the proof.
The following result is a straight consequence of Lemma 4.13.
Lemma 4.14. Let C, C ′ be two connected components ofω ∆ . LetB(x, R) be a ball of C and T ′ be a hole of C ′ which does not containB(x, R). Then, Lemma 4.15. Let T and T ′ be two holes respectively of two connected components C and C ′ ofω ∆ . If T ⊂ T ′ and T ′ ⊂ T then d(T, T ′ ) ≥ 2R 0 .
Proof. Let T and T ′ be two holes satisfying the assumption of the lemma. We denote by x and y two points belonging respectively to the closure of T and T ′ such that d(T, T ′ ) = |x− y|. The point x (respectively y) belongs to the boundary of two balls B(z, R) and B(z ′ , R ′ ) of C (respectively B(w, r) and B(w ′ , r ′ ) of C ′ ). An analysis, as in the proof of Lemma 4.13, shows that the distance |x − y| is minimal in the situation where R = R ′ = r = r ′ = R 0 and z, z ′ , w and w ′ form a parallelogram with length side 2R 0 . Then the points x and y are at the middle of two opposite sides and the result follows.