Nicolas Chenavier
email: nicolas.chenavier@etu.univ-rouen.fr

A general study of extremes of stationary tessellations with examples

Keywords: Random tessellations, extreme values, order statistics, dependency graph, Poisson approximation, Voronoi flower, Poisson point process, Gauss-Poisson point process AMS 2010 Subject Classifications: 60D05 . 60G70 . 60G55 . 60F05 . 62G32

Let m be a random tessellation in R d , d ≥ 1, observed in a bounded Borel subset W and f (•) be a measurable function defined on the set of convex bodies. A point z(C), called the nucleus of C, is associated with each cell C of m. Applying f (•) to all the cells of m, we investigate the order statistics of f (C) over all cells C ∈ m with nucleus in Wρ = ρ 1/d W when ρ goes to infinity. Under a strong mixing property and a local condition on m and f (•), we show a general theorem which reduces the study of the order statistics to the random variable f (C), where C is the typical cell of m. The proof is deduced from a Poisson approximation on a dependency graph via the Chen-Stein method. We obtain that the point process (ρ

, where aρ > 0 and bρ are two suitable functions depending on ρ, converges to a non-homogeneous Poisson point process. Several applications of the general theorem are derived in the particular setting of Poisson-Voronoi and Poisson-Delaunay tessellations and for different functions f (•) such as the inradius, the circumradius, the area, the volume of the Voronoi flower and the distance to the farthest neighbor.

Introduction

A tessellation of R d , d ≥ 1, endowed with its Euclidean norm | • |, is a countable collection of nonempty compact subsets, called cells, with disjoint interiors which subdivides the space and such that the number of cells intersecting any bounded subset of R d is finite. The set T of tessellations is endowed with the σ-algebra generated by the sets m ∈ T, C∈m ∂C ∩ K = ∅ where ∂K is the boundary of K for any compact set K in R d . By a random tessellation m, we mean a random variable with values in T. It is said to be stationary if its distribution is invariant under translations of the cells. For a complete account on random tessellations, we refer to the books [START_REF] Schneider | Stochastic and integral geometry. Probability and its Applications[END_REF][START_REF] Stoyan | Stochastic Geometry and its applications[END_REF] and the survey [START_REF] Calka | Tessellations. In New perspectives in stochastic geometry[END_REF].

Given a fixed realization of m, we associate with each cell C ∈ m in a deterministic way a point z(C), which is called the nucleus of the cell, such that z(C + x) = z(C) + x for all x ∈ R d . To describe the mean behaviour of the tessellation, the notions of intensity and typical cell are introduced as follows. Let B be a Borel subset of R d such that λ d (B) ∈ (0, ∞), where λ d is the d-dimensional Lebesgue measure. The intensity γ of the tessellation is defined as γ = 1 λ d (B) • E [#{C ∈ m, z(C) ∈ B}] and we assume that γ ∈ (0, ∞). Since m is stationary, γ is independent of B and we suppose, without loss of generality, that γ = 1. The typical cell C is a random polytope whose the distribution is given by

E[f (C)] = 1 λ d (B) • E     C∈m, z(C)∈B f (C -z(C))     (1.1)
for all f : K d → R bounded measurable function on the set of convex bodies K d , i.e. convex compact sets, endowed with the Hausdorff topology. We are interested in the following problem: only a part of the tessellation is observed in the window W ρ = ρ 1/d W , where W is a bounded Borel subset of R d , i.e. included in a cube C (W ) , and such that λ d (W ) = 0. Let f : K d → R be a translation invariant measurable function, i.e. f (C + x) = f (C) for all C ∈ K d and x ∈ R d . For all r ∈ N * , we denote by M (r) f,Wρ the r-th order statistic of f over the cells C ∈ m such that z(C) ∈ W ρ . We have chosen the convention to call the r order statistics the r largest values. When r = 1, the 1-st order statistic is denoted by M f,Wρ , i.e.

M f,Wρ = M (1) f,Wρ = max C∈m, z(C)∈Wρ f (C).
In this paper, we investigate the limit behaviour of M (r) f,Wρ when ρ goes to infinity. The study of extremes could describe the regularity of the tessellation (e.g. presence of elongated cells). For instance, in finite element method, the quality of the approximation depends on some consistency measurements over the partition, see e.g. [START_REF] Ju | Adaptive finite element methods for elliptic PDEs based on conforming centroidal Voronoi-Delaunay triangulations[END_REF]. Another potential application field is statistics of point processes. The key idea would be to identify a point process from the extremes of a tessellation induced by the point process.

To the best of our knowledge, one of the first works on extreme values in stochastic geometry is due to Penrose. In chapters 6,7 and 8 in [START_REF] Penrose | of Oxford Studies in Probability[END_REF], he investigates the maximum and minimum degrees of random geometric graphs. More recently, Schulte and Thäle [START_REF] Schulte | The scaling limit of Poisson-driven order statistics with applications in geometric probability[END_REF] establish a theorem to derive the order statistics of a functional f k (x 1 , . . . , x k ) of k points on a homogeneous Poisson point process. Nevertheless, their approach cannot be applied to our problem. Indeed, studying extremes of the tessellation requires to use functionals which depend on the whole point process of nuclei and not only on a fixed number of points. Besides, we consider any translation invariant measurable function f (•) and we restrict our investigation to a certain kind of random tessellation satisfying a strong mixing property (Condition (FRC), p. 3). Our general theorem (Theorem 1, p. 3) is followed by numerous examples, with rates of convergence, in the particular setting of Poisson-Voronoi and Poisson-Delaunay tessellations. This improves in particular some extremes that are investigated in [START_REF] Calka | Extreme values for characteristic radii of a Poisson-Voronoi tessellation[END_REF]. Before stating our main theorems, we need some preliminaries which contain notation and conditions on the random tessellation.

Preliminaries. Let C (W ) be a cube in R d containing W . We partition C (W ) ρ = ρ 1/d C (W ) into a set V ρ of N ρ sub-cubes of equal size, such that N . With a slight abuse of notation, we identify a cube with its index. Let us define a distance between sub-cubes i and j as d(i, j) = max 1≤r≤d {|i r -j r |}. Moreover, if A, B are two sets of sub-cubes, we let d(A, B) = min i∈A,j∈B d(i, j). For each i ∈ V ρ , we denote by

M f,i = max C∈m, z(C)∈i∩Wρ f (C).
(1.2)

When {C ∈ m, z(C) ∈ i ∩ W ρ } is empty, we take M f,i = -∞.
Let us consider a threshold v ρ depending on ρ. Studying the order statistics amounts to investigate the number of exceedance cells of v ρ denoted by

U ρ (v ρ ) = C∈m, z(C)∈Wρ 1 f (C)>vρ .
(1.3)

Since f is translation invariant, the mean of this random variable is, according to (1.1):

E [U ρ (v ρ )] = λ d (W ρ ) • P (f (C) > v ρ ) . (1.4)
We assume the following condition which is referred as the typical cell property (TCP):

Condition (TCP): the mean number of exceedance cells converges to a limit denoted by τ ≥ 0, i.e.

λ d (W ρ ) • P (f (C) > v ρ ) -→ ρ→∞ τ.
Moreover, we denote by G 1 (ρ) the rate of convergence, i.e.

G 1 (ρ) = |λ d (W ρ ) • P(f (C) > v ρ ) -τ | . (1.5)
We assume also a (global) condition of R-dependence associated with m and f which is referred as the finite range condition (FRC).

Condition (FRC): there exists an integer R and an event A ρ with P (A ρ ) -→ ρ→∞ 1 such that, conditional on A ρ , the σ-algebras σ{M f,i , i ∈ A} and σ{M f,i , i ∈ B} are independent when d(A, B) > R.

Finally, in order to present our first theorem, we introduce a second function defined as

G 2 (ρ) = N ρ E      (C1,C2) = ∈m 2 , z(C 1 ),z(C 2 )∈Cρ 1 f (C1)>vρ,f (C2)>vρ      , ( 1.6)
where

C ρ = 0, (2R + 1) • λ d (W ρ ) 1/d N -1/d ρ d (1.7)
and where (C 1 , C 2 ) = ∈ m 2 means that (C 1 , C 2 ) is a pair of distinct cells of m.

Order statistics. We are now prepared to present our first theorem.

Theorem 1. Let m be a stationary random tessellation of intensity 1 such that Conditions (TCP) and (FRC) hold and let r ∈ N * be fixed. Then

P(M (r) f,Wρ ≤ v ρ ) -e -τ r-1 k=0 τ k k! = O G 1 (ρ) + G 2 (ρ) + N -1 ρ + P(A c ρ ) , (1.8)
where φ(ρ) = O(ψ(ρ)) means that φ(ρ)/ψ(ρ) is bounded as ρ goes to infinity.

To derive useful applications, we assume a second condition on the random tessellation referred as the local correlation condition (LCC).

Condition (LCC): the function G 2 (ρ) converges to 0 as ρ goes to infinity. This (local) condition means that with high probability two neighboring cells are not simultaneously exceedances. Under this assumption, we obtain the following result: Corollary 1. Let m be a stationary random tessellation of intensity 1 such that Conditions (TCP), (FRC) and (LCC) hold and let r ∈ N * be fixed. Then

P(M (r) f,Wρ ≤ v ρ ) -→ ρ→∞ e -τ r-1 k=0 τ k k! with rate of convergence O G 1 (ρ) + G 2 (ρ) + N -1 ρ + P(A c ρ ) .
Remark 1. When Condition (LCC) does not hold, we can show that lim ρ→∞ P M f,Wρ ≤ v ρ = e -θτ for some constant θ ∈ [0, 1], provided that the limit of P M f,Wρ ≤ v ρ exists. Its proof relies notably on the adaptation to our setting of several arguments included in [START_REF] Leadbetter | Extremes and local dependence in stationary sequences[END_REF]. According to Leadbetter, we say that θ is the extremal index. In a future work, we hope to develop a general method to estimate this quantity.

Theorem 1 could be extended in several directions. First, its result remains true for random tessellations satisfying some β-mixing property (see examples of Voronoi tessellations given in [START_REF] Heinrich | Normal approximation for some mean-value estimates of absolutely regular tessellations[END_REF]). Moreover, it can be generalized to marked point processes satisfying three properties: two conditions similar to Conditions (FRC) and (LCC), and one condition on the mark distribution. In particular, we could investigate extremes of Boolean models with bounded grains. When the random tessellation is ergodic with respect to the group of translations of R d , the order statistics are asymptotically independent of the choice of nuclei z(•). Indeed, they only depend on the asymptotic behaviour of G 1 (ρ) and the typical cell C itself does not depend on the set of nuclei thanks to the ergodic theorem (see [START_REF] Daley | An introduction to the theory of point processes[END_REF], p. 339). This will be the case for examples that we deal with. Moreover, we notice that the order statistics do not depend on the shape of the window W . Actually, a method similar to Proposition 3 of [START_REF] Calka | Extreme values for characteristic radii of a Poisson-Voronoi tessellation[END_REF] shows that the contribution of boundary cells is negligible. Besides, we do not know if our rate of convergence is optimal. The technique that we use is similar to the method employed by Penrose and Yukich [START_REF] Penrose | Normal approximation in geometric probability[END_REF]. Nevertheless, their rate of convergence is sub-optimal (e.g. better estimates can be found in [START_REF] Last | Normal approximation on Poisson spaces: Mehler's formula, second order Poincaré inequalities and stabilization[END_REF]). So we suppose that the improvement of the results might be achieved by using other techniques, different from ours.

As mentioned above, Conditions (FRC) and (LCC) concern global and local properties of the tessellation respectively. In fact, there exists an analogy between Conditions (FRC) and (LCC) and Conditions D(u n ) and D (u n ) of Leadbetter [START_REF] Leadbetter | On extreme values in stationary sequences[END_REF] respectively. The general theory of extreme values deals with sequences [START_REF] Hsing | On the extreme order statistics for a stationary sequence[END_REF] or random fields [START_REF] Leadbetter | On extreme values in stationary random fields[END_REF], see also the reference books [START_REF] De Haan | Extreme value theory. An introduction[END_REF] and [START_REF] Resnick | Extreme values, regular variation, and point processes, volume 4 of Applied Probability[END_REF]. Unfortunately, we are unable to apply it in our setting. Indeed, the set of random variables that we consider is not a discrete random field in a classical meaning. More precisely, the process {M f,i } i∈Vρ is a triangular array indexed by N d and the process {f (C x )} x∈R d is not a Gaussian continuous random field, where C x is the cell of the tessellation containing x.

Point process of exceedances. Concretely, the threshold is often of the form

v ρ = v ρ (t) = a ρ t + b ρ , t ∈ R with a ρ > 0.
In this case, we can be more specific about the joint distributions of the order statistics. Before stating our second theorem, we need some preliminaries. We denote by τ (t) ∈ [0, +∞], t ∈ R, the limit of λ d (W ρ ) • P(f (C) > v ρ (t)) as ρ goes to infinity and by * x = inf{t ∈ R, τ (t) < ∞} and x * = sup{t ∈ R, τ (t) > 0} the lower and upper endpoints of τ (•). Since a ρ is positive, the function τ (•) is not increasing so that τ (•) is finite on ( * x, x * ].

Under Conditions (FRC) and (LCC), we consider the random collection

Φ ρ = ρ -1/d z(C), a -1 ρ (f (C) -b ρ ) , C ∈ m and z(C) ∈ W ρ ⊂ W × R.
Moreover, we consider a Poisson point process Φ ⊂ W × ( * x, x * ], with intensity measure ν given by

ν(B × (s, t]) = E [#Φ ∩ (B × (s, t])] = λ d (B) λ d (W ) • (τ (s) -τ (t))
for all Borel subset B ⊂ W and all segment (s, t] ⊂ ( * x, x * ]. We then obtain the following limit theorem.

Theorem 2. Let m be a stationary random tessellation of intensity 1 such that Conditions (TCP), (FRC) and (LCC) hold for each

v ρ = v ρ (t) = a ρ t + b ρ , t ∈ R.
⊂ W × ( * x, x * ] with ν(∂B i ) = 0, 1 ≤ i ≤ k (#Φ ρ ∩ B 1 , . . . , #Φ ρ ∩ B k ) D -→ (#Φ ∩ B 1 , . . . , #Φ ∩ B k ) ,
where ∂B i denotes the boundary of B i .

This result suggests that the largest order statistics can be seen as points of a (non homogeneous) Poisson point process. Theorem 2 gives their joint distributions so that Corollary 1 is a particular case of the latter when k = 1 and B = W × (t, x * ]. For a wider panorama on results of the point process of exceedances associated with the extremes of a sequence of non independent random variables, we refer to chapter 5 in [START_REF] Leadbetter | Extremes and related properties of random sequences and processes[END_REF]. When W = C (W ) = [0, 1] d and when τ (•) is not constant, the function τ (•) belongs to either the Fréchet, the Gumbel or the Weibull family. This fact is a rewriting of the proof of Theorem 4.1 in [START_REF] Leadbetter | On extreme values in stationary random fields[END_REF].

The paper is organized as follows. In Section 2, we show how to reduce our problem to the study of extreme values on a dependency graph. We use a result of [START_REF] Arratia | Poisson approximation and the Chen-Stein method[END_REF] to derive an estimate of exceedances by a Poisson distribution. We then deduce Theorems 1 and 2 from a discretization of W into sub-cubes. Sections 3, 4 and 5 are devoted to numerous applications on Delaunay and Voronoi random tessellations. We investigate the asymptotic behaviours with the rates of convergence of:

• the minimum of circumradii of a Poisson-Delaunay tessellation in any dimension and the maximum and minimum of the areas in the planar case (Section 3),

• the minimum of distances to the farthest neighboring nucleus and the minimum of the volume of flowers for a Poisson-Voronoi tessellation (Section 4),

• the maximum of inradii for a Voronoi tessellation induced by a Gauss-Poisson process (Section 5).

For each tessellation and each characteristic, we need to find a suitable threshold v ρ and to check Condition (LCC) which requires some delicate geometric estimates.

In the rest of the paper, c or c denotes a generic constant which does not depend on ρ but may depend on other quantities. The term v ρ = v ρ (t) denotes a generic function of t, depending on ρ, which is specified in Sections 3, 4 and 5.

Proofs of Theorems 1 and 2.1 Extreme values on a dependency graph and proof of Theorem 1

We first outline the methodology of the proof of Theorem 1 with some additional notation. A classical method in extreme value theory is to investigate the exceedances. We consider two random variables that are the number of exceedance cells U ρ (v ρ ), introduced in (1.3), and the number of exceedance cubes U Vρ (v ρ ) defined as

U Vρ (v ρ ) = i∈Vρ 1 M f,i >vρ , ( 2.1)
where V ρ and M f,i are introduced in the preliminaries, p. 2. We denote by µ ρ the mean of U Vρ (v ρ ), i.e.

µ ρ = E U Vρ (v ρ ) = i∈Vρ P (M f,i > v ρ ) . (2.
2)

The proof of Theorem 1 is divided into the three following lemmas.

Lemma 1. Under the same assumptions as in Theorem 1, we get for all r ∈ N *

P (U ρ (v ρ ) ≤ r -1) -P U Vρ (v ρ ) ≤ r -1 ≤ 2 • G 2 (ρ). (2.3)
The above lemma is a consequence of Condition (LCC).

Lemma 2. Let µ ρ be as in (2.2). Under the hypothesis of Theorem 1, we get for all r ∈ N *

P(U Vρ (v ρ ) ≤ r -1) -e -µρ r-1 k=0 µ k ρ k! = O G 2 (ρ) + N -1 ρ + P(A c ρ ) . (2.4)
The derivation of Lemma 2 constitutes the major part of the proof of Theorem 1. It means that the number of exceedance cubes is approximately a Poisson random variable. The fundamental concept to prove this lemma is that of a dependency graph. We first establish a Poisson approximation on the number of exceedances on such graph and we show how our problem can be reduced to this setting. Finally, the following result gives an estimate for µ ρ . Lemma 3. Let µ ρ as in (2.2). Under the assumptions as in Theorem 1, we get

|µ ρ -τ | ≤ G 1 (ρ) + G 2 (ρ).
Proof of Theorem 1. Since M (r) f,Wρ is lower than v ρ if and only if U ρ (v ρ ) ≤ r -1, we deduce Theorem 1 from the three lemmas above and the fact that the function x → e -x r-1 k=0

x k k! is Lipschitz.
In the rest of this subsection, we proceed as follows. We first establish the Poisson approximation on a dependency graph (Proposition 3) and we deduce from it Lemma 2. The key idea is to apply Conditions (TCP) and (FRC). Then we prove Lemmas 1 and 3.

Extreme values on a dependency graph. Given a collection of real random variables X i , i ∈ V (not necessarily stationary), we say that the graph G = (V, E) is a dependency graph for X i if for any pair of disjoint sets A 1 , A 2 ⊂ V such that no edge in E has one endpoint in A 1 and the other in A 2 , the σ-fields σ(X i , i ∈ A 1 ) and σ(X i , i ∈ A 2 ) are mutually independent. Introduced by Petrovskaya and Leontovitch in [START_REF] Petrovskaya | The central limit theorem for a sequence of random variables with a slowly growing number of dependences[END_REF], this concept was applied by Baldi and Rinott (e.g. [START_REF] Baldi | On normal approximations of distributions in terms of dependency graphs[END_REF]) to obtain central limit theorems and normal approximations. Furthermore, Arratia et al. give a Poisson approximation of a sum of (non independent) Bernoulli random variables indexed by a countable set (see Theorem 1 in [START_REF] Arratia | Poisson approximation and the Chen-Stein method[END_REF]). We write their result in our context to approximate the number of exceedances on a dependency graph by a Poisson random variable.

We denote by |V |, D and J ⊂ R, the number of vertices of G, its maximal degree and a finite union of disjoint intervals respectively. Let U V (J) be

U V (J) = i∈V 1 X i ∈J
and p i = P(X i ∈ J), p ij = P(X i ∈ J, X j ∈ J) for all i ∈ V and j ∈ V (i) -{i}, where V (i) is the set of neighbors of i, i.e.

V (i) = {j ∈ V, (i, j) ∈ E} ∪ {i}.
Let us consider a Poisson random variable Z of mean

µ J = E[Z] = E[U V (J)] = i∈V P (X i ∈ J) .
The Chen-Stein method can be applied to approximate the number of occurrences of dependent events by a Poisson random variable (e.g. [START_REF] Arratia | Poisson approximation and the Chen-Stein method[END_REF]). In particular, this is a powerful tool to derive some results in extreme value theory for a sequence of real random variables (e.g. [START_REF] Smith | Extreme value theory for dependent sequences via the Stein-Chen method of Poisson approximation[END_REF]). We write below a slightly modified version of Theorem 1 of [START_REF] Arratia | Poisson approximation and the Chen-Stein method[END_REF] to derive an upper bound for the total variation distance between U V (J) and its Poisson approximation Z for a dependency graph. Proposition 3. [START_REF] Arratia | Poisson approximation and the Chen-Stein method[END_REF]) Let p(V ) = sup i∈V p i and q(V ) 2 = sup (i,j)∈E p ij . Then

sup A⊂N |P(U V (J) ∈ A) -P(Z ∈ A)| ≤ 2D • |V | • p(V ) 2 + q(V ) 2 .
(2.5)

In particular, for all r ∈ N * , we get

P(U V (J) ≤ r -1) -e -µ J r-1 k=0 µ k J k! ≤ 2D • |V | • p(V ) 2 + q(V ) 2 . ( 2.6)
Proof of Proposition 3. The upper bound (2.6) is a direct consequence of (2.5). From Theorem 1 of [START_REF] Arratia | Poisson approximation and the Chen-Stein method[END_REF], we get

sup A⊂N |P(U V (J) ∈ A) -P(Z ∈ A)| ≤ 2(b 1 + b 2 + b 3 ), (2.7)
where

b 1 = i∈V j∈V (i) p i p j , b 2 = i∈V i =j∈V (i) p ij and b 3 = i∈V E [|P (X i ∈ J|σ(X j : j ∈ V (i))) -p i |] . Since |V (i)| ≤ D + 1, we obtain b 1 ≤ |V | • D • p(V ) 2 and b 2 ≤ |V | • D • q(V ) 2 .
Moreover, using the fact that if j ∈ V (i), the random variable X j is independent of X i , we get b 3 = 0. We then deduce (2.5) from (2.7).

The technique of dependency graphs, combined with a discretization technique, has been used on several occasions to derive central limit theorems in geometric probability (see e.g. [START_REF] Avram | On central limit theorem in geometrical probability[END_REF]). In the same spirit, we derive Lemma 2 from Proposition 3. We need first to explain how we construct the dependency graph from our random tessellation.

Construction of the dependency graph. We define a graph G ρ = (V ρ , E ρ ) as follows. The set V ρ consists of the sub-cubes i (|V ρ | = N ρ ) which cover W ρ and an edge (i, j) ∈ E ρ if d(i, j) ≤ R, where R is introduced in Condition (FRC). The maximal degree D ρ of this graph satisfies

D ρ ≤ (2R + 1) d . (2.8)
For all i ∈ V ρ , we define the random variable X i as X i = M f,i . From Condition (FRC), conditional on A ρ , the graph G ρ is a dependency graph for (M f,i ) i∈Vρ .

Proof of Lemma 2. We apply Proposition 3 to X i = M f,i and J = (v ρ , ∞). It is enough to derive upper bounds for P(M f,i > v ρ |A ρ ) and

P(M f,i > v ρ , M f,j > v ρ |A ρ ). According to (1.
2), we get

P(M f,i > v ρ ) = P     C∈m, z(C)∈i∩Wρ {f (C) > v ρ }     ≤ E     C∈m, z(C)∈i∩Wρ 1 f (C)>vρ     .
Since f is translation invariant and ) , we deduce from (1.1) that

λ d (i) = 1 Nρ λ d C (W ) ρ = c(W ) • 1 Nρ λ d (W ρ ), where c(W ) = λ d (C (W ) ) λ d (W
P(M f,i > v ρ ) ≤ c(W ) • 1 N ρ λ d (W ρ ) • P(f (C) > v ρ ).
(2.9)

Using the inequalities

P(M f,i > v ρ |A ρ ) ≤ P(M f,i > v ρ )/P(A ρ ) and λ d (W ρ ) • P(f (C) > v ρ ) ≤ G 1 (ρ) + τ , where G 1 (ρ)
is defined in (1.5), we obtain

p i := P(M f,i > v ρ |A ρ ) ≤ c(W ) • G 1 (ρ) + τ P(A ρ )N ρ . (2.10)
Moreover, for any i ∈ V ρ and j ∈ V ρ (i) -{i}, we get

P(M f,i > v ρ , M f,j > v ρ ) = P     C1∈m, z(C 1 )∈i∩Wρ C2∈m, z(C 2 )∈j∩Wρ {f (C 1 ) > v ρ , f (C 2 ) > v ρ }     ≤ E      (C1,C2) = ∈m 2 z(C 1 ),z(C 2 )∈Vρ(i) 1 f (C1)>vρ,f (C2)>vρ      , ( 2.11)
where

(C 1 , C 2 ) = ∈ m 2 means that (C 1 , C 2
) is a pair of distinct cells. By a slight abuse of notation, we will write in the rest of the paper V ρ (i) for the union of the sub-cubes j∈Vρ(i) j.

Besides, the set of neighbors V ρ (i) can be re-written as

V ρ (i) = {j ∈ V ρ , d(i, j) ≤ R}. Hence V ρ (i) is a convex union of disjoint sub-cubes of volume λ d (W ρ )/N ρ ,
which are at most (2R + 1) d , and can be included in the cube C ρ defined in (1.7) up to a translation. Since f is translation invariant, we obtain

E      (C1,C2) = ∈m 2 z(C 1 ),z(C 2 )∈Vρ(i) 1 f (C1)>vρ,f (C2)>vρ      ≤ G 2 (ρ) N ρ .
(2.12)

Using the fact that

P(M f,i > v ρ , M f,j > v ρ |A ρ ) ≤ P(M f,i > v ρ , M f,j > v ρ )/P(A ρ ) we deduce from (2.11) that p ij := P(M f,i > v ρ , M f,j > v ρ |A ρ ) ≤ G 2 (ρ) P(A ρ )N ρ . (2.13)
Applying (2.6) to the conditional probability given A ρ , we deduce from (2.8), (2.10), (2.13) and from the fact

|V ρ | = N ρ that P(U Vρ (v ρ ) ≤ r -1|A ρ ) -e -µρ r-1 k=0 µ k ρ k! ≤ 2(2R + 1) d P(A ρ ) 2 • c(W ) 2 • (G 1 (ρ) + τ ) 2 N ρ + P(A ρ )G 2 (ρ) .
The rate of convergence (2.4) results directly from the previous upper bound and the fact that P(A ρ ) and G 1 (ρ) converge respectively to 1 and 0 according to Conditions (TCP) and (FRC).

Proof of Lemma 1. Let us notice that Lemma 1 is trivial when r = 1. More generally, for all r ∈ N * , we have

P (U ρ (v ρ ) ≤ r -1) -P U Vρ (v ρ ) ≤ r -1 ≤ 2P U ρ (v ρ ) = U Vρ (v ρ ) . (2.14)
According to (1.3) and (2.1), the above random variables differ if and only if there are at least two exceedances in the same sub-cube i, i.e. (2.12). This shows that

P U ρ (v ρ ) = U Vρ (v ρ ) = P      i∈Vρ (C1,C2) = ∈m 2 , z(C 1 ),z(C 2 )∈i∩Wρ {f (C 1 ) > v ρ , f (C 2 ) > v ρ }      ≤ i∈Vρ E      (C1,C2) = ∈m 2 z(C 1 ),z(C 2 )∈i∩Wρ 1 f (C1)>vρ,f (C2)>vρ      . Since |V ρ | = N ρ , the right-hand side is bounded by G 2 (ρ) thanks to
P U ρ (v ρ ) = U Vρ (v ρ ) ≤ G 2 (ρ) and consequently we deduce (2.3) from (2.

14).

Proof of Lemma 3. From (2.2) and the triangle inequality, we get

|µ ρ -τ | ≤ |E [U ρ (v ρ )] -τ | + E U ρ (v ρ ) -U Vρ (v ρ ) ,
where U ρ (v ρ ) ≥ U Vρ (v ρ ) a.s. According to (1.4) and (1.5), we obtain that

|µ ρ -τ | ≤ G 1 (ρ) + E U ρ (v ρ ) -U Vρ (v ρ ) . (2.15)
To give an upper bound for the second term of the right-hand side of (2.15), we use the fact that the family V ρ covers W ρ . Intuitively, the number of exceedance sub-cubes U Vρ (v ρ ) can be approximated by the number of exceedance cells U ρ (v ρ ) when G 2 (ρ) is negligible. We prove this fact below. From (1.3) and (2.1), we obtain a.s. that

U ρ (v ρ ) -U Vρ (v ρ ) = i∈Vρ C∈m, z(C)∈i∩Wρ 1 f (C)>vρ -1 M f,i >vρ = i∈Vρ     C∈m, z(C)∈i∩Wρ 1 f (C)>vρ -1     1 M f,i >vρ ≤ i∈Vρ (C1,C2) = ∈m 2 z(C 1 ),z(C 2 )∈i∩Wρ 1 f (C1)>vρ,f (C2)>vρ .
(2.16)

The last inequality comes from the fact that if there is 0 or 1 exceedance cell inside a sub-cube i, the sums over

C ∈ m, z(C) ∈ i ∩ W ρ and (C 1 , C 2 ) = ∈ m 2 , z(C 1 ), z(C 2 ) ∈ i ∩ W ρ are null. Otherwise, if the number of exceedances is k ≥ 2, we use the fact k -1 ≤ k(k-1)
Taking expectations in (2.16) and using the fact that the mean of the right-hand side of (2.16) is bounded by G 2 (ρ) as in the proof of Lemma 1, we get

E U ρ (v ρ ) -U Vρ (v ρ ) ≤ G 2 (ρ).
(2.17)

It results from the previous inequality and (2.15

) that |µ ρ -τ | is bounded by G 1 (ρ) + G 2 (ρ).
Remark 2. As mentioned in Section 1, p. 4, Theorem 1 can be extended to more general tessellations. Indeed, Condition (FRC) can be replaced by the following property: "β(ρ

) := N ρ sup i∈Vρ β (σ(M f,i ), σ(M f,j , d(i, j) > R))
converges to 0 as ρ goes to infinity", where β(A, B) is the β-mixing coefficient between two σ-algebras A and B. This comes from the fact that b 3 , which appears in (2.7), can be bounded by β(ρ).

Proof of Theorem 2

By Kallenberg's theorem (see Proposition 3.22, p. 156 in [START_REF] Resnick | Extreme values, regular variation, and point processes, volume 4 of Applied Probability[END_REF], see also the proof of Theorem 2.1.2 in [START_REF] De Haan | Extreme value theory. An introduction[END_REF]) it is enough to check that:

• for all Borel subset B ⊂ W and * x < s ≤ t ≤ x * E [#Φ ρ ∩ (B × (s, t])] -→ ρ→∞ E [#Φ ∩ (B × (s, t])] , (2.18)
• for all

P = L l=1 B (l) × (s l , t l ],
where B (l) is the intersection of W and a rectangular solid in C (W ) and

* x < s l ≤ t l ≤ x * P (#Φ ρ ∩ P = 0) -→ ρ→∞ P (#Φ ∩ P = 0) . (2.19)
Proof of (2.18). From (1.1), we have

E [#Φ ρ ∩ (B × (s, t])] = E     C∈m, z(C)∈Bρ 1 aρs+bρ<f (C)≤aρt+bρ     = λ d (B ρ ) • (P (f (C) > a ρ s + b ρ ) -P (f (C) > a ρ t + b ρ )) ,
where

B ρ = ρ 1/d B. Since λ d (B ρ ) = λ d (B) λ d (W ) • λ d (W ρ ) and λ d (W ρ ) • P (f (C) > v ρ (t)) converges to τ (t) for all t ∈ R, we get E [#Φ ρ ∩ (B × (s, t])] -→ ρ→∞ λ d (B) λ d (W ) • (τ (s) -τ (t)) = E [#Φ ∩ (B × (s, t])] (2.20)
and consequently we obtain (2.18).

Proof of (2.19). We can write P as a disjoint union of strips, i.e.

P = L l=1 B (l) × J (l) (2.21)
such that the Borel subsets B (l) ⊂ W are disjoint and such that J (l) is a finite union of half-open intervals, 1 ≤ l ≤ L.

The following lemma shows that it is enough to investigate the case where P is a strip.

Lemma 4.

Let P be as in (2.21). Under the hypothesis of Theorem 2, we have

P (#Φ ρ ∩ P = 0) - L l=1 P #Φ ρ ∩ (B (l) × J (l) ) = 0 -→ ρ→∞ 0.
The proof of Lemma 4 is postponed at the end of the subsection. Thanks to Lemma 4, we can assume that P, defined in (2.21), is only a strip P = B × J, where J is a finite union of half-open intervals and B ⊂ W . Without loss of generality, we can assume that these intervals are disjoint, i.e.

J = k j=1 (s j , t j ] (2.22) with * x < s j ≤ t j ≤ x * and t j ≤ s j+1 , 1 ≤ j ≤ k.
In the same spirit as in the proof of Theorem 1, we introduce the two random variables

U ρ (B × J) = #Φ ρ ∩ (B × J) = C∈m, z(C)∈Bρ 1 a -1 ρ (f (C)-bρ)∈J and U Vρ (B × J) = i∈Vρ 1 a -1 ρ (M f,i (B)-bρ)∈J , ( 2.23)
where

M f,i (B) = max C∈m, z(C)∈i∩Bρ f (C).
In particular,

U ρ (W × (s, ∞)) = U ρ (v ρ (s)) and U Vρ (W × (s, ∞)) = U Vρ (v ρ (s)), where U ρ (v ρ (s)) and U Vρ (v ρ (s)) are defined in (1.
3) and (2.1). We denote by µ ρ (B × J) the mean of U Vρ (B × J), i.e.

µ ρ (B × J) = E U Vρ (B × J) = i∈Vρ P a -1 ρ (M f,i (B) -b ρ ) ∈ J .
As in the proof of Theorem 1, we subdivide the proof into three steps. More precisely, we show that

P (U ρ (B × J) = 0) -P U Vρ (B × J) = 0 -→ ρ→∞ 0, (2.24a)
P U Vρ (B × J) = 0 -e -µρ(B×J) -→ ρ→∞ 0, (2.24b
)

µ ρ (B × J) -→ ρ→∞ ν(B × J). (2.24c)
Notice that the convergences (2.24a), (2.24b) and (2.24c) are generalisations of Lemmas 1, 2 and 3 respectively. For the proof of (2.24a), it is enough to show that P U ρ (B × J) = U Vρ (B × J) converges to 0 as ρ goes to infinity.

Since U ρ (B × J) ≥ U Vρ (B × J) for all Borel subsets, we have

P U ρ (B × J) = U Vρ (B × J) ≤ k j=1 P U ρ (B × (s j , t j ]) = U Vρ (B × (s j , t j ]) ≤ k j=1 P U ρ (W × (s j , ∞)) = U Vρ (W × (s j , ∞)) = k j=1 P U ρ (v ρ (s j )) = U Vρ(vρ(sj )) .
The proof of Lemma 1 shows that the right-hand side converges to 0. Secondly, we prove (2.24b). In the same spirit as in the proof of Lemma 2, we apply Proposition 3 conditional on A ρ to

X i = a -1 ρ (M f,i (B) -b ρ ) and J = k j=1 (s j , t j ]. Let i ∈ V ρ and j ∈ V ρ (i) -{i}. Since M f,i (B) ≤ M f,i , we get p i = P a -1 ρ (M f,i (B) -b ρ ) ∈ J|A ρ ≤ P (M f,i (B) > v ρ (s 1 )|A ρ ) = O(N -1 ρ )
according to (2.9). Moreover (2.13). We deduce (2.24b) from the previous inequalities and Proposition 3.

p ij = P a -1 ρ (M f,i (B) -b ρ ) ∈ J, a -1 ρ (M f,j (B) -b ρ ) ∈ J|A ρ ≤ P (M f,i (B) > v ρ (s 1 ), M f,i (B) > v ρ (s 1 )|A ρ ) = O G 2 (ρ) • N -1 ρ according to
Thirdly, we prove (2.24c). According to (2.22) and (2.23), we have a.s.

U ρ (B × J) = k j=1 #Φ ρ ∩ (B × (s j , t j ]).
Taking expectations in the previous equality, we deduce from (2.20) that

E [U ρ (B × J)] -→ ρ→∞ λ d (B) λ d (W ) k j=1 (τ (s j ) -τ (t j )) = ν(B × J). (2.25) Moreover E [U ρ (B × J)] -µ ρ (B × J) = E U ρ (B × J) -U Vρ (B × J) ≤ E U ρ (v ρ (s 1 )) -U Vρ (v ρ (s 1 )) (2.26)
converges to 0 according to (2.17). We deduce (2.24c) from (2.25) and (2.26). Finally, according to (2.24a), (2.24b), (2.24c) and the fact that U ρ (B × J) = #Φ ρ ∩ (B × J), we deduce that

P (#Φ ρ ∩ (B × J) = 0) -→ ρ→∞ e -ν(N ×J) = P (#Φ ∩ (B × J) = 0)
and consequently we obtain (2.19).

The end of the subsection is devoted to the proof of Lemma 4.

Proof of Lemma 4.

Let P = L l=1 B (l) × J (l) and B (l) = B l ∩ W such that the rectangular solids B l ⊂ C (W ) are disjoint. We denote respectively by V ρ (B (l) ), S ρ (B (l) ) and V • ρ (B (l) ) the sets        V ρ (B (l) ) = {i ∈ V ρ , i ∩ B l = ∅}, S ρ (B (l) ) = {i ∈ V ρ , i ∩ ∂B l = ∅}, V • ρ (B (l) ) = {i ∈ V ρ (B (l) ), d(i, S ρ (B (l) )) > R}.
Finally, we consider the random variable

U V • ρ (B (l) × J (l) ) ≤ U ρ (B (l) × J (l)
) defined as

U V • ρ (B (l) × J (l) ) = i∈V • ρ (B (l) ) 1 a -1 ρ (M f,i (B (l) )-bρ)∈J (l) . Let 1 ≤ l ≤ L be fixed. Since B l is a rectangular solid in C (W ) which is covered with at most N ρ sub-cubes i, we have #S ρ (B (l) ) ≤ c • N (d-1)/d ρ
. This shows that

P U V • ρ (B (l) × J (l) ) = U Vρ (B (l) × J (l) ) ≤ #S ρ (B (l) ) • P (M f,i > v ρ ) = O N -1/d
ρ according to (2.9) and Condition (TCP). Thanks to (2.24a), we deduce that

P U ρ (B (l) × J (l) ) = 0 -P U V • ρ (B (l) × J (l) ) = 0 -→ ρ→∞ 0. (2.27)
Moreover, conditional on A ρ , the random variables

U V • ρ (B (l) × J (l) ) are independent since U V • ρ (B (l) × J (l) ) is σ M f,i , i ∈ V • ρ (B (l) ) measurable and d V • ρ (B (l) ), V • ρ (B (l ) ) > R for all 1 ≤ l = l ≤ L.
In particular, we get

P L l=1 U V • ρ (B (l) × J (l) ) = 0 A ρ = L l=1 P U V • ρ (B (l) × J (l) ) = 0 A ρ .
Lemma 4 is a consequence of the previous equality, the convergence (2.27) and the fact that

P (#Φ ρ ∩ P = 0) = P L l=1 {U ρ (B (l) × J (l)) ) = 0} .
Remark 3. The inequalities appearing in (1.5), (1.6) and Theorem 1 have to be reversed when we deal with the r smallest values. This fact will be extensively used in the rest of the paper.

In the three following sections, we apply Theorem 1 to derive the asymptotic behaviours of the order statistics for different geometrical characteristics and random tessellations. For aesthetic reasons, we only investigate maxima and minima for the particular case W = C (W ) = [0, 1] d keeping in mind that these results can be generalized to order statistics and to any bounded set with λ d (W ) = 0. Up to a normalization, all the thresholds v ρ can be written as v ρ = v ρ (t) = a ρ t + b ρ (excepted in Section 5) so that Theorem 2 is also available.

Extreme Values of a Poisson-Delaunay tessellation

Before applying Theorem 1 to different geometrical characteristics of a Poisson-Delaunay tessellation, we introduce some notation and preliminaries.

Notation. For all z ∈ R d and r ≥ 0, we denote by B(z, r) and S(z, r) the ball and the sphere of radius r centered in z. When z = 0 and r = 1, we denote by S d-1 = S(0, 1) the unit sphere and κ d = λ d (B(0, 1)). Let C be a simplex in R d , i.e. the convex hull of (d + 1) points. When the extremal points of C are affinely independent, we denote by B(C), S(C), z(C) and R(C) the circumball, the circumsphere, the circumcenter and the circumradius of C respectively. Otherwise, we take

B(C) = S(C) = {0}, z(C) = 0 and R(C) = 0.
Let k be an integer and x 1 , . . . , x k be k points in R d and let f be a function defined on K d with values on some measurable space.

• For all r ≥ 0, we denote by rx 1:k the k-tuple (rx 1 , . . . , rx k ) and by {rx 1:k } the set of points {rx 1 , . . . , rx k }.

• When k = d + 1, we denote by ∆(x 1:d+1 ) the convex hull of x 1 , . . . , x d+1 and f (x 1:d+1 ) = f (∆(x 1:d+1 )).

• If k ≤ d+1 and if {y k+1:d+1 } = {y k+1 , . . . , y d+1 } is a set of d+1-k points in R d , we denote by ∆(x 1:k , y k+1:d+1 ) the convex hull of x 1 , . . . , x k , y k+1 , . . . , y d+1 and f (x 1:k , y k+1:d+1 ) = f (∆ (x 1:k , y k+1:d+1 )). In particular, if k = 0, we take f (x 1:0 , y 1:d+1 ) = f (y 1:d+1 ).

• Finally, we denote by dσ(u) and dσ(u

1:d+1 ) = dσ(u 1 ) • • • dσ(u d+1
) the uniform distribution over S d-1 and the product measure respectively.

Preliminaries. Let χ be a locally finite subset of R d such that each subset of size n < d+1 are affinely independent and no d + 2 points lie on a sphere. If d + 1 points x 1 , . . . , x d+1 of χ lie on a ball that contains no point of χ in its interior, then the convex hull of x 1 , . . . , x d+1 is called a cell. The set of such cells defines a partition of R d into simplices and such partition is called the Delaunay tessellation (see [START_REF] Schneider | Stochastic and integral geometry. Probability and its Applications[END_REF], p. 478). Such model is the key ingredient of the first algorithm for computing the minimum spanning tree [START_REF] Shamos | Closest-point problems[END_REF]. It is extensively used in medical image segmentation [START_REF] Spanel | Delaunay-Based Vector Segmentation of Volumetric Medical Images[END_REF], in finite element method to build meshes [START_REF] Ju | Adaptive finite element methods for elliptic PDEs based on conforming centroidal Voronoi-Delaunay triangulations[END_REF] and is a powerful tool for reconstructing a 3D set from a discrete point set [START_REF] Schaap | DTFE: the Delaunay Tessellation Field Estimator[END_REF].

When χ = X is a Poisson point process, we speak about Poisson-Delaunay tessellation denoted by m P DT . For each cell C ∈ m P DT which is a.s. a simplex, the nucleus z(C) is defined as the circumcenter of C. The relation between the intensity γ of m P DT and the intensity γ X of the underlying Poisson point process is given by (see Section 7 in [START_REF] Møller | Random tessellations in R d[END_REF])

γ = β -1 d • γ X ,
where

β d = (d 3 + d 2 )Γ d 2 2 Γ d d+1 2 Γ d 2 +1 2 Γ d d+2 2 2 d+1 π d-1 2 . ( 3.1)
In particular, we have

β 1 = 1, β 2 = 1 2 and β 3 = 35 24π 2 .
To be in the framework of Theorem 1, we assume without loss of generality that γ = 1, i.e. γ X = β d . Moreover, we partition the window

W ρ = ρ 1/d [0, 1] d into N ρ := ρ 2 log ρ 1/d d sub-cubes i ∈ V ρ ,
where x denotes the integer part of x for any x ∈ R. We think that this choice for N ρ is quasi-optimal for the rate of convergence in Theorem 1. Indeed, according to (1.8), this rate is of order G 1 (ρ) + G 2 (ρ) + N -1 ρ + P A c ρ , where G 1 (ρ) is independent of N ρ . Besides, according to (3.4), we have P A c ρ ≤ N ρ e -ρ/Nρ . This implies that N ρ e -ρ/Nρ has to converge to 0 so that N ρ = O ρ log ρ . Moreover, the larger N ρ is, the smaller N -1 ρ and G 2 (ρ) are. If N ρ ∼ ρ→∞ a•ρ log ρ for some constant a < 1, the choice of a does not affect the order of G 2 (ρ). Hence, N ρ seems to be optimal when it minimizes N -1 ρ + N ρ e -ρ/Nρ i.e. when N ρ is close to ρ 2 log ρ . To apply Theorem 1, we first check Condition (FRC) for any measurable function f : K d → R. To do it, we define the event A ρ as

A ρ = i∈Vρ {X ∩ i = ∅}. (3.2) Lemma 5. Let f : K d → R be a measurable function. Then Condition (FRC) is satisfied for R = 2 • √ d + 1 and for the event A ρ defined in (3.2).
Proof of Lemma 5. We use the same arguments as in the proof of Proposition 3 in [START_REF] Avram | On central limit theorem in geometrical probability[END_REF]. Let i ∈ V ρ be a sub-cube in W ρ and let C ∈ m P DT such that z(C) ∈ i. Since a (d + 1)-tuple of points of X is a Delaunay cell if and only if its circumball contains no point in its interior, we have R(C) = min x∈X {|z(C) -x|}. Moreover, conditional on A ρ , there exists a point x 0 in X ∩ i. In particular, we have |z(C) -

x 0 | ≤ √ d • c ρ
, where c ρ is the length of the sides of each sub-cube. Consequently, we obtain

R(C) ≤ √ d • c ρ . (3.3)
This shows that the circumsphere d(i,A) < D} and {i, d(i, B) < D} are disjoint and because X ∩ {i, d(i, A) < D} and X ∩ {i, d(i, B) < D} are independent, the σ-algebras σ(M f,i , i ∈ A) and

S(C) of C is included in V ρ (i, D) := {j ∈ V ρ , d(i, j) ≤ D}, where D = √ d + 1. Indeed if not, there exists a point y ∈ S(C) such that y is in a sub-cube j with d(i, j) ≥ D + 1. This shows that |y -z(C)| > ( √ d + 1) • c ρ and contradicts (3.3) since R(C) = |y -z(C)|. Since S(C) is included in V ρ (i, D) for any cell C ∈ m P DT such that z(C) ∈ i, this shows that M f,i is σ(X ∩ V ρ (i, D)) measurable. Because d(A, B) > 2D implies that {i,
σ(M f,i , i ∈ B) are independent, yielding R = 2D = 2 • √ d + 1 .
Moreover the probability of the event A ρ converges to 1. Indeed, since X is a Poisson point process, we get

P(A c ρ ) = P   i∈Vρ {X ∩ i = ∅}   ≤ N ρ e -ρ/Nρ = O (log ρ) -1 × ρ -1 . (3.4)
Besides, the distribution function of the typical cell can be made explicit. Indeed, let f : K d → R be a translation invariant function on the set of convex bodies. An integral representation of f (C), due to Miles [START_REF] Miles | A synopsis of "Poisson flats in Euclidean spaces[END_REF] (the proof can also be found in Theorem 10.4.4. of [START_REF] Schneider | Stochastic and integral geometry. Probability and its Applications[END_REF]), is given by

E [f (C)] = δ d • ∞ 0 (S d-1 ) d+1 r d 2 -1 e -δ d r d λ d (u 1:d+1 )f (ru 1:d+1 )drdσ(u 1:d+1 ) with δ d = (d+1)β d and δ d = κ d β d . (3.5)
For practical reasons, we write below a generic lemma which gives an integral representation of the function G 2 (•) defined in (1.6). To do it, we introduce some notation. As defined in (1.6)

, G 2 (•) involves two different simplices ∆ 1 , ∆ 2 such that f (∆ i ) > v ρ and z(∆ i ) ∈ C ρ , i = 1, 2. The intersection of these cells is a k-dimensional simplex with 0 ≤ k ≤ d -1.
Translating the circumcenter of the cell which has the largest circumradius say ∆ 1 at the origin, the cells can be written as ∆ 1 = ∆(ru 1:d+1 ) and ∆ 2 = ∆(ru 1:k , y k+1:d+1 ) with r ≥ 0, u 1 , . . . , u d+1 ∈ S d-1 and y k+1 , . . . , y d+1 ∈ R d . We consider the two properties: We are now prepared to state the generic lemma.

Lemma 6. Let m P DT be a Poisson-Delaunay tessellation of intensity

γ = 1. Then G 2 (ρ) = 2 • d k=0 G 2,k (ρ), (3.9)
where Proof of Lemma 6. This will be sketched since it is in the same spirit as in the proof of (3.5). Considering that the intersection of the two Delaunay cells ∆ 1 , ∆ 2 which appear in (1.6) is a k-dimensional simplex with 0 ≤ k ≤ d and assuming that R(∆ 1 ) ≥ R(∆ 2 ), we have

G 2,k (ρ) = ρ ∞ 0 (S d-1 ) d+1 (R d ) d+1-k g 2,k (ρ,
G 2 (ρ) = 2 d k=0 E (x1,...,x d+1 ) = ∈X d+1 (y 1 ,...,y k ) = ∈X k 1 f (x d+1 )>vρ 1 f (x 1:k ,y k+1:d+1 )>vρ 1 R(x 1:d+1 )≥R(x 1:k ,y k+1:d+1 ) × 1 X∩B (∪) (x 1:d+1 ,y k+1:d+1 )\{x 1:d+1 }∪{y k+1:d+1 } = ∅ ,
where B (∪) (x 1:d+1 , y k+1:d+1 ) = B(x 1:d+1 ) ∪ B(x 1:k , y k+1:d+1 ). It results from Slivnyak-Mecke formula (see e.g. Theorem 3.3.5 in [START_REF] Schneider | Stochastic and integral geometry. Probability and its Applications[END_REF]) that

G 2 (ρ) = 2 d k=0 (R d ) d+1 (R d ) d+1-k 1 f (x d+1 )>vρ 1 f (x 1:k ,y k+1:d+1 )>vρ 1 R(x 1:d+1 )≥R(x 1:k ,y k+1:d+1 ) × P #X ∩ B (∪) (x 1:d+1 , y k+1:d+1 ) = 0 dy k+1:d+1 dx 1:d+1 .
We conclude the proof of Lemma 6 by noting that #X ∩ B (∪) (x 1:d+1 , y k+1:d+1 ) is Poisson distributed with mean β d λ d B (∪) (x 1:d+1 , y k+1:d+1 ) and using for all y k+1 , . . . , y d+1 the (Blaschke-Petkantschin type) change of variables

φ 1 :R + × R d × (S d-1 ) d+1 -→ (R d ) d+1 (r, z, u 1:d+1 ) -→ x 1:d+1 with x i = z + ru i , (3.12)
where the Jacobian is |Dφ 1 (r, z,

u 1:d+1 )| = r d 2 -1 λ d (u 1:d+1 ).
In (3.6a), we have assumed that R (ru 1:k , y k+1:d+1 ) is less than r. This property will be needed in Sections 3.2 and 3.3 to bound λ d+1-k (E k,r,u 1:d+1 ) with respect to r (see Lemmas 8 and 9) but not in Section 3.1 since we could bound λ d+1-k (E k,r,u 1:d+1 ) there with respect to v ρ (as defined in (3.15)) instead of r.

Minimum of the circumradii

In this subsection, we investigate the minimum of circumradii, i.e.

R min,P DT (ρ) = min

R(C).

The asymptotic behaviour of R min,P DT (ρ) is given in the following proposition.

Proposition 4. Let m P DT be a Poisson-Delaunay tessellation of intensity

γ = 1 in R d , d ≥ 2. Then for all t ≥ 0 P α 1/d d,1 ρ 1/d R min,P DT (ρ) d ≥ t -e -t d = O ρ -1/d , ( 3.13)
where

α d,1 = δ d d d! = (κ d β d ) d d! = 1 d! •   (d 3 + d 2 )Γ d 2 2 Γ d d+1 2 π 1/2 2 d+1 Γ d 2 +1 2 Γ d+1 d+2 2   d .
Notice that the asymptotic behaviour of the maximum of circumradii has been investigated in [START_REF] Calka | Extreme values for characteristic radii of a Poisson-Voronoi tessellation[END_REF].

Proof of Proposition 4. First, we give the asymptotic behaviour of the distribution function of R(C). According to (3.5), the random variable R(C) d is Gamma distributed with parameters d 2 , δ -1 d . Thanks to consecutive integrations by parts, this proves that

P(R(C) < v) = ∞ i=d 1 i! (δ d v d ) i e -δ d v d (3.14)
for all v ≥ 0. A Taylor approximation of the right-hand side when v goes to 0 shows that

|P(R(C) < v) -α d,1 • v d 2 | is of order v d 2 +d
. Hence, taking for all t ≥ 0

v ρ = v ρ (t) = α -1 d,1 ρ -1 1/d 2 t 1/d , ( 3.15)
we obtain

G 1 (ρ) = |ρP(R(C) < v ρ ) -t d | = O ρ -1/d . ( 3.16)
To calculate an upper bound of G 2 (ρ), it is enough to give a suitable upper bound for G 2,k (ρ), 0 ≤ k ≤ d according to Lemma 6. Bounding the exponential in (3.11) by 1 (a suitable estimate when considering small cells) and λ d (u 1:d+1 ) by a constant, we deduce for all r ∈ R + , u 1:d+1 ∈ (S d-1 ) d+1 and y k+1:d+1 ∈ (R d ) d+1-k that g 2,k (ρ, r, u 1:d+1 , y k+1:d+1 ) ≤ c • r d 2 -1 1 r<vρ 1 E k,r,u 1:d+1 (y k+1:d+1 ).

(3.17)

When k = 0, we bound 1 E0,r,u 1:d+1 (y 1:d+1 ) by 1 R(y 1:d+1 )<r • 1 z(y 1:d+1 )∈Cρ . Integrating the right-hand side of (3.17) and taking the same change of variables as in (3.12), i.e. y i = z + r u i , 1 ≤ i ≤ d + 1, we deduce from (3.10) and (3.15) that

G 2,0 (ρ) ≤ c • ρλ d (C ρ ) vρ 0 r d 2 -1 r 0 r d 2 -1 dr dr = O log ρ • ρ -1 . (3.18) When 1 ≤ k ≤ d, we use the fact that R(ru 1:k , y k+1:d+1 ) < r implies y i ∈ B(ru 1 , 2r) for all k + 1 ≤ i ≤ d + 1. Bounding 1 E k,
r,u 1:d+1 (y k+1:d+1 ) by 1 y k+1 ,...,y d+1 ∈B(ru1,2r) and integrating (3.17), we deduce from (3.10) that

G 2,k (ρ) ≤ c • ρ vρ 0 S d-1 (R d ) d+1
r d 2 -1 1 y k+1 ,...,y d+1 ∈B(ru1,2r) dy k+1:d+1 dσ(u 1 )dr When d = 1, the same method shows that 2ρR min,P DT (ρ) converges to a standard exponential distribution. The only difference with the case d ≥ 2 is that the rate of convergence is log ρ • ρ -1 (and not ρ -1 ) according to (3.18) and (3.19).

≤ c • ρ vρ 0 r d 2 -1 × r d(d+1-k) dr = O ρ -(d+1-k)/d . ( 3
Let us remark that a slightly weaker version of Proposition 4 in R d could have been deduced from a theorem due to Schulte and Thäle (see Theorem 1.1 in [START_REF] Schulte | The scaling limit of Poisson-driven order statistics with applications in geometric probability[END_REF]). It comes from the fact that R min,P DT (ρ) can be written as a minimum of a U -statistic. More precisely R min,P DT (ρ) = min

x 1:d+1 ∈X d+1 z(x 1:d+1 )∈Wρ , R(x 1:d+1 ).
Indeed, if a simplex induced by a set of (d+1) distinct points x 1:d+1 of X minimizes the circumradius, it is necessarily a Delaunay cell: otherwise, the circumball B(x 1:d+1 ) contains a point of X in its interior which contradicts the minimality of R(x 1:d+1 ). Nevertheless, the rate of convergence O ρ -1/d of Proposition 4 is more accurate than the rate deduced from Theorem 1.1. in [START_REF] Schulte | The scaling limit of Poisson-driven order statistics with applications in geometric probability[END_REF] since the latter is of order O ρ -1/2d . To the best of our knowledge, the convergence of the point process provided by Theorem 2 applied to the circumscribed radius of Delaunay cells is new.

Maximum of the areas, d = 2

Here and in the subsequent subsection, we investigate the extremes of the areas of a planar Poisson-Delaunay tessellation of intensity 1. The extension to higher dimension would be intricate since the integral formula for the distribution function of the volume of the typical cell becomes intractable. The intensity of the underlying Poisson point process is

γ X = β 2 = 1 2 . ( 3.20)
In this subsection, we investigate the maximum of the areas, i.e.

A max,P DT (ρ) = max

C∈m P DT z(C)∈Wρ , λ 2 (C).
The following proposition shows that A max,P DT (ρ) is of order log ρ.

Proposition 5. Let m P DT be a Poisson-Delaunay tessellation of intensity

γ = 1 in R 2 . Then for all t ∈ R P α 2 A max,P DT (ρ) -log 3 2 ρ ≤ t -e -e -t = O (1/ log ρ) , (3.21)
where

α 2 = 2π 3 √ 3 .
Proof of Proposition 5. Thanks to (3.5), the distribution function of λ 2 (C) can be made explicit. Indeed, when d = 2, an integral representation of P (λ 2 (C) > v) due to Rathie (see (3.2) in [START_REF] Rathie | On the volume distribution of the typical Poisson-Delaunay cell[END_REF]) is

P(λ 2 (C) > v) = 6 π ∞ α2β2v xK 2 1/6 (x)dx, (3.22)
where K 1/6 (•) denotes the modified Bessel function of order 1/6. When x goes to infinity, a Taylor approximation of K 1/6 (x) is given by (see Formula 9.7.2, p. 378 in [START_REF] Abramowitz | Handbook of mathematical functions with formulas, graphs, and mathematical tables[END_REF])

K 1/6 (x) = π 2x e -x 1 + O 1 x . (3.23)
We deduce from (3.20), (3.22) and (3.23) that for v large enough

P (λ 2 (C) > v) - 3 2 e -α2v ≤ c • ∞ 1 2 α2v e -2x x dx ≤ c • e -α2v v . (3.24)
Taking for all t ∈ R

v ρ = v ρ (t) = 1 α 2 log 3 2 ρ + t , (3.25)
we obtain from (3.24) that

G 1 (ρ) = |ρP (λ 2 (C) > v ρ ) -e -t | = O (1/ log ρ) .
In the rest of the proof, we give a suitable upper bound for G 2 (ρ). Taking f (•) = λ 2 (•) in (3.11) and using the fact that λ 2 (ru 1:3 ) = r 2 λ 2 (u 1:3 ) and λ 2 (u 1:3 ) ≤ c, we have

g 2,k (ρ, r, u 1:3 , y k+1:3 ) ≤ c • r 3 e -1 2 λ (∪) d (r,u 1:k ,y k+1:3 ) 1 r 2 λ2(u1:3)>vρ 1 E k,r,u 1:3 (y k+1:3 ) (3.26)
for all k = 0, 1, 2. To bound g 2,k (•), the key idea is to derive a suitable lower bound for the area of the union of two disks (see Figure 1 (a)). This is provided in the following fundamental lemma. Moreover, the maximal area of a triangle inscribed in a ball of radius R is 3 √ 3 4 R 2 which is the area of an equilateral triangle. In particular, we have λ 2 (x 1:3 ) ≤ 3 √ 3 4 R 2 . This together with (3.28) implies that

Lemma 7. Let {x 1:3 } = {x 1 , x 2 , x 3 } and {x 1:3 } = {x 1 , x 2 , x 3 } be two sets of three points in R 2 such that x i ∈ B(x 1:3 ) and x j ∈ B(x 1:3 ) for all i, j = 1, 2, 3. Let us assume that R := R(x 1:3 ) ≥ R(x 1:3 ). Then λ 2 (B(x 1:3 ) ∪ B(x 1:3 )) ≥ π 2 -1 R 2 + λ 2 (x 1:3 ) + λ 2 (x 1:3 ). ( 3
λ 2 (B(x 1:3 ) ∪ B(x 1:3 )) ≥ π - 3 √ 3 4 R 2 + λ 2 (x 1:3 ) + λ 2 (x 1:3 ) ≥ π 2 -1 R 2 + λ 2 (x 1:3 ) + λ 2 (x 1:3 ).
If B(x 1:3 ) ∩ B(x 1:3 ) has nonempty interior, the intersection of the circumspheres induced by the points x 1:3 and x 1:3 is reduced to two points, say p 1 , p 2 ∈ R 2 . Let us denote by L the affine line (p 1 , p 2 ) and H -(respectively H + ) the half plane delimited by L and containing (respectively not containing) the circumcenter z(x 1:3 ). Since

x i ∈ B(x 1:3 ) and x j ∈ B(x 1:3 ), i, j = 1, 2, 3, the triangle ∆(x 1:3 ) is included in H + . Hence λ 2 (B(x 1:3 ) ∪ B(x 1:3 )) = λ 2 (B(x 1:3 ) ∪ B(x 1:3 )) ∩ H -+ λ 2 (B(x 1:3 ) ∪ B(x 1:3 )) ∩ H + ≥ λ 2 (B(x 1:3 ) ∩ H -) + λ 2 (x 1:3 ). (3.29)
In the rest of the proof, we provide a suitable lower bound for λ 2 (B(x 1:3 ) ∩ H -). To do it, we denote by θ ∈ [0, 2π] the angle ∠p 1 z(x 1:3 )p 2 . Actually θ ∈ [0, π]: this comes from the fact that λ 2 (B(x We discuss below two cases depending on θ. When θ ∈ [0, 2π/3], we deduce from (3.30) that

1:3 ) ∩ H -) ≥ π 2 R 2 since R := R(x 1:3 ) ≥ R(x 1:3 ). The area of the cap B(x 1:3 ) ∩ H -is given by λ 2 (B(x 1:3 ) ∩ H -) = π - 1 2 (θ -sin θ) R 2 . ( 3
λ 2 (B(x 1:3 ) ∩ H -) ≥ 2π 3 + √ 3 4 R 2 . (3.31) Since λ 2 (x 1:3 ) is less than 3 √ 3 4 R 2 , we deduce from (3.31) that λ 2 (B(x 1:3 ) ∩ H -) ≥ λ 2 (x 1:3 ) + 2π 3 - √ 3 2 R 2 ≥ λ 2 (x 1:3 ) + π 2 -1 R 2 . (3.32)
In that case, the inequality (3.27) results from (3.29) and (3.32). When θ ∈ [2π/3, π], with a standard method of geometry, we can show that the maximal area M (θ) of a triangle inscribed in B(x

1:3 ) ∩ H -is M (θ) = sin θ 2 + 1 2 sin θ R 2 . (3.33)
Actually, the triangle which maximizes the area is isoscele with central angles π -θ/2, π -θ/2 and θ (see Figure 1 (b)). We obtain from (3.30) and (3.33) that

λ 2 (B(x 1:3 ) ∩ H -) ≥ M (θ) + π 2 -1 R 2 + π 2 + 1 - 1 2 θ + sin θ 2 R 2 .
The last term of the right-hand side is a decreasing function on [0, π]. Its minimum equals 0 at θ = π, i.e.

π 2 + 1 - 1 2 θ + sin θ 2 ≥ 0 for all θ ∈ [0, π]. This shows that λ 2 (B(x 1:3 ) ∩ H -) ≥ M (θ) + π 2 -1 R 2 . (3.34)
The inequality (3.27) results from (3.29), (3.34) and the fact that M (θ) ≥ λ 2 (x 1:3 ).

We can now derive an upper bound for g 2,k (•) for all k = 0, 1, 2. Indeed, if y k+1:3 ∈ E k,r,u1:3 , where E k,r,u1:3 has been defined in (3.7), the sets of points {x 1:3 } = {ru 1:3 } and {x 1:3 } = {ru 1:k , y k+1:3 } satisfy the assumptions of Lemma 7 since R(ru 1:3 ) = r and R(ru 1:3 ) ≥ R(ru 1:k , y k+1:3 ). Using the fact that B(ru 1:3 ) = B(0, r), λ 2 (ru 1:3 ) > v ρ and λ 2 (ru 1:k , y k+1:3 ) > v ρ , we deduce from (3.8), (3.26) and (3.27) that

g 2,k (ρ, r, u 1:3 , y k+1:3 ) ≤ c • r 3 e -1 2 (( π 2 -1)r 2 +2vρ) 1 r 2 λ2(u1:3)>vρ 1 E k,r,u 1:3 (y k+1:3 ). Since 3 √ 3 4 r 2 ≥ r 2 λ 2 (u 1:3 ) and α 2 = 2π 3 √ 3 , we deduce from (3.25) that r 2 λ 2 (u 1:3 ) > v ρ implies r > (2 (log ρ + c ) /π) 1/2
, where c = log(3/2) + t. Integrating the right-hand side with respect to y k+1:3 , we obtain

G 2,k (ρ) ≤ c • ρ ∞ (2(log ρ+c )/π) 1/2 (S 1 ) 3 r 3 e -1 2 (( π 2 -1)r 2 +2vρ) × λ 2(3-k) (E k,r,u1:3 )dσ(u 1:3 )dr. (3.35)
The following lemma gives a uniform upper bound for λ 2(3-k) (E k,r,u1:3 ).

Lemma 8. Let u 1:3 ∈ (S 1 ) 3 and r > (2 (log ρ + c ) /π) 1/2 . Then for ρ large enough

λ 2(3-k) (E k,r,u1:3 ) ≤ c • r 2(3-k) . (3.36)
Proof of Lemma 8. We discuss the three cases k = 0, 1, 2.

If k = 2, we show that E 2,r,u1:3 is included in a ball of radius r up to a multiplicative constant and centered at 0. Let y 3 be in E 2,r,u1:3 . From the triangle inequality, we have

|y 3 | ≤ |y 3 -z(ru 1:2 , y 3 )| + |z(ru 1:2 , y 3 )| ≤ r + diam(C ρ ).
(3.37)

The last inequality comes from the fact that |y 3 -z(ru 1:2 , y 3 )| is the circumradius of ∆(ru 1:2 , y 3 ), which is less than r, and the fact that z(ru

1:2 ) ∈ C ρ . Moreover diam(C ρ ) ≤ c • (log ρ) 1/2 ≤ c • r, (3.38)
where the last inequality holds for ρ large enough since r > (2 (log ρ + c ) /π) 1/2 converges to infinity as ρ goes to infinity. We deduce from (3.37) and (3.38) that |y 3 | ≤ c • r. This shows that E 2,r,u1:3 ⊂ B(0, c • r). In particular, we get λ 2 (E 2,r,u1:3 ) ≤ c • r 2 . If k = 1 or k = 0, proceeding along the same lines as in the case k = 2, we show that E k,r,u1:3 ⊂ B(0, c • r) 3-k and consequently we get

λ 2(3-k) (E k,r,u1:3 ) ≤ c • r 2(3-k) .
We can now derive an upper bound for G 2,k (ρ). Indeed, integrating u 1:3 over (S 1 ) 3 , we deduce from (3.35) and (3.36) that

G 2,k (ρ) ≤ c • ρ ∞ (2(log ρ+c )/π) 1/2 r 9-k e -1 2 (( π 2 -1)r 2 +2vρ) dr.
Integrating the right-hand side, we obtain from (3.25) that

G 2,k (ρ) ≤ c • (log ρ) 8-2k ρ (π+2-3 √ 3)/2π = O (log ρ) 8 ρ - (3.39)
with = -π -2 + 3 √ 3 > 0. Proposition 2 results from (3.39), Lemma 6 and Theorem 1.

Lemma 7 provides the main tool of the proof. We can note that inequality (3.27) is obvious when we replace

π 2 -1 by a constant α ≤ π -3 √ 3
2 . Indeed, if ∆(x 1:3 ) and ∆(x 1:3 ) are two triangles with R :

= R(x 1:3 ) ≥ R(x 1:3 ), a trivial inequality is λ 2 (B(x 1:3 ) ∪ B(x 1:3 )) ≥ πR 2 . Consequently λ 2 (B(x 1:3 ) ∪ B(x 1:3 )) ≥ π - 3 √ 3 2 R 2 + λ 2 (x 1:3 ) + λ 2 (x 1:3 )
since λ 2 (x 1:3 ) and λ 2 (x 1:3 ) are less than 3

√ 3 4 R 2 .
Nevertheless, the previous lower bound is not enough to guarantee that G 2,k (ρ) converges to 0. The important fact in Lemma 7 is that we consider the more precise constant

π 2 -1 > π -3 √ 3
2 . Another remark deals with the shape of the cell maximizing the area. Indeed, the maximum of circumradii of a planar Poisson-Delaunay tessellation, denoted by R max,P DT (ρ), satisfies (see (2c) in [START_REF] Calka | Extreme values for characteristic radii of a Poisson-Voronoi tessellation[END_REF]):

P π 2 R max,P DT (ρ) 2 -log (ρ log ρ) ≤ t -→ ρ→∞ e -e -t
for all t ∈ R. This shows that the expectation of 3 √ 3

4 R max,P DT (ρ) 2 , which is the area of an equilateral triangle of circumradius R max,P DT (ρ), is of order 3 √ 3 2π log ρ. According to (3.21), this is also the order of E [A max,P DT (ρ)]. It seems that the shape of the cell maximizing the area tends to that of an equilateral triangle. This fact can be connected to the D.G. Kendall's conjecture and to the work of Hug and Schneider in [START_REF] Hug | Large cells in Poisson-Delaunay tessellations[END_REF].

Minimum of the areas, d = 2

In our third example, we calculate the asymptotic behaviour of the minimum of the areas of the cells of a Poisson-Delaunay tessellation (of intensity 1) in R 2 , i.e.

A min,P DT (ρ) = min

where

α 3 = 2 -2/3 • 3 -1/2 • 5 -1 • π 2/3 • Γ(1/6) 2 .
In [START_REF] Schulte | The scaling limit of Poisson-driven order statistics with applications in geometric probability[END_REF], Schulte and Thäle investigate the behaviour of the smallest area S ρ of all triangles that can be formed by three points of the Poisson point process, i.e.

S ρ = min x1:3∈X 3 z(x 1:3 )∈Wρ , λ 2 (x 1:3 ).
The asymptotic behaviour of S ρ (see Theorem 2.5. in [START_REF] Schulte | The scaling limit of Poisson-driven order statistics with applications in geometric probability[END_REF]) is given by P (ρS ρ ≥ t) -→ ρ→∞ e -βt , where β is a constant which can be made explicit. The previous limit compared to (3.40) shows that the smallest area of the Delaunay cells is much larger than the smallest area of all triangles.

Proof of Proposition 6. First, we calculate the asymptotic behaviour of the distribution function of λ 2 (C) (see (3.22)). A Taylor expansion of K 1/6 (x) as x goes to 0 is given by (see Formula 9.6.9, p. 375 in [START_REF] Abramowitz | Handbook of mathematical functions with formulas, graphs, and mathematical tables[END_REF])

K 1/6 (x) = 2 -5/6 Γ (1/6) x -1/6 + o(x -1/6 ).

22

This together with (3.20) and (3.22) shows that

P (λ 2 (C) < v) = 6 π • 2 -5/3 Γ (1/6) 2 α2β2v 0 x 2/3 + o x 2/3 dx = α 3 • v 5/3 + o v 5/3 . ( 3.41)
Taking for all t ≥ 0

v ρ = v ρ (t) = (α -1 3 ρ -1 ) 3/5 t (3.42)
we obtain

G 1 (ρ) = |ρP (λ 2 (C) < v ρ ) -t 5/3 | -→ ρ→∞ 0. (3.43)
We investigate below the rate of convergence of G 2 (ρ). Taking f (ru 1:3 ) = r 2 λ 2 (u 1:3 ) and using the fact that λ 2 (B(0, r) ∪ B(ru 1:k , y k+1:3 )) is greater than πr 2 , for all k = 0, 1, 2, we have

g 2,k (ρ, r, u 1:3 , y k+1:3 ) ≤ r 3 e -1 2 πr 2 λ 2 (u 1:3 )1 r 2 λ2(u1:3)<vρ 1 E k,r,u 1:3
according to (3.8) and (3.11). Integrating with respect to y 1:3 , this gives

G 2,k (ρ) ≤ ρ ∞ 0 (S 1 ) 3 r 3 e -1 2 πr 2 λ 2 (u 1:3 )λ 2(3-k) (E k,r,u1:3 )1 r 2 λ2(u1:3)<vρ dσ(u 1:3 )dr. (3.44)
As in the proof of Proposition 5, we derive a suitable upper bound for the volume of E k,r,u1:3 .

Lemma 9.

Let u 1:3 ∈ (S 1 ) 3 such that u 1 = u 2 and r > 0. Then

λ 2 (E 2,r,u1:3 ) ≤ c • v ρ |u 1 -u 2 | -1 , (3.45a) λ 4 (E 1,r,u1:3 ) ≤ c • r 2 v ρ , (3.45b) λ 6 (E 0,r,u1:3 ) ≤ c • log ρ • r 2 v ρ . (3.45c)
Proof of Lemma 9. Let y 3 be in E 2,r,u1:3 . Since R(ru 1:2 , y 3 ) is less than r, we have |y 3 -ru 1 | ≤ 2R(ru 1:2 , y 3 ) ≤ 2r.

In particular, we obtain

|y 3 | ≤ 3r. (3.46)
Moreover, the area of the triangle ∆(ru 1:2 , y 3 ) is given by

λ 2 (ru 1:2 , y 3 ) = 1 2 r|u 1 -u 2 | • δ(y 3 , L(ru 1 , ru 2 )), (3.47)
where L(ru 1 , ru 2 ) is the line through p 1 = ru 1 , p 2 = ru 2 and where δ(y 3 , L(ru 1 , ru 2 )) denotes the distance between this line and the point y 3 . Since λ 2 (ru 1:2 , y 3 ) < v ρ , it results from (3.47) that

δ(y 3 , L(ru 1 , ru 2 )) ≤ 2v ρ r|u 1 -u 2 | . ( 3.48)
The inequalities (3.46) and (3.48) show that E 2,r,u1:3 is included in the intersection of a ball of radius 3r and a strip of width 4vρ r|u1-u2| , i.e.

λ 2 (E 2,r,u1:3 ) ≤ 6r × 4v ρ r|u 1 -u 2 | = c • v ρ |u 1 -u 2 | -1 .
Secondly, we bound λ 4 (E 1,r,u1:3 ). Taking the (spherical coordinates type) change of variables

φ 2 : R + × S 1 → R 2 , (s , u 2 ) → y 2 = ru 1 + s u 2 with Jacobian |Dφ 2 (s , u 2 )| = s , we obtain λ 4 (E 1,r,u1:3 ) ≤ 2r 0 S 1 R 2 s 1 λ2(ru1,ru1+s u 2 ,y3)<vρ 1 R(ru1,ru1+s u 2 ,y3)≤r dy 3 dσ(u 2 )ds . (3.49)
The positive number s is integrated on (0, 2r]. Indeed, the inequality R(ru

1 , ru 1 + s u 2 , y 3 ) ≤ r implies that s = |(ru 1 + s u 2 ) -ru 1 | ≤ 2r.
Proceeding along the same lines as in the proof of (3.45a), we show that y 3 belongs to the ball B(0, 3r) and a strip of width 4vρ s . Integrating (3.49) with respect to y 3 , we deduce that

λ 4 (E 1,r,u1:3 ) ≤ 24 2r 0 S 1 v ρ rdσ(u 2 )ds = c • r 2 v ρ .
Finally, we bound λ 6 (E 0,r,u1:3 ). Taking the same change of variables as in (3.12), we have

λ 6 (E 0,r,u1:3 ) ≤ (R 2 ) 3 1 z(y1:3)∈Cρ 1 R(y1:3)<r 1 λ2(y1:3)<vρ dy 1:3 = Cρ r 0 (S 1 ) 3 r 3 λ 2 (u 1:3 )1 r 2 λ2(u 1 ,u 2 ,u 3 )<vρ dσ(u 1:3 )dr dz . Bounding r 3 λ 2 (u 1 , u 2 , u 3 ) by r v ρ and integrating with respect to z ∈ C ρ , r ∈ [0, r] and u 1:3 ∈ (S 1 ) 3 , we show that λ 6 (E 0,r,u1:3 ) is less than c • λ 2 (C ρ )r 2 v ρ with λ 2 (C ρ ) ≤ c • log ρ.
We can now derive a suitable upper bound for G 2,k (ρ). Indeed, if k = 0, we deduce from (3.44) and (3.45c) that

G 2,0 (ρ) ≤ c • log ρ • ρv ρ ∞ 0 (S 1 ) 3 r 5 e -1 2 πr 2 λ 2 (u 1:3 )1 r 2 λ2(u1:3)<vρ dσ(u 1:3 )dr ≤ c • log ρ • ρv 2 ρ ∞ 0 (S 1 ) 3 r 3 e -1
2 πr 2 dσ(u 1:3 )dr.

First, we notice that the integral of the right-hand side is bounded. Besides, (3.42) shows that G 2,0 (ρ) is less than c • log ρ • ρ -1/5 . In the same spirit, when k = 1, we obtain that G 2,1 (ρ) ≤ c • ρ -1/5 according to (3.44) and (3.45b). Hence

G 2,0 (ρ) = O log ρ • ρ -1/5 and G 2,1 (ρ) = O ρ -1/5 . (3.50)
Finally, if k = 2, we deduce from (3.44) and (3.45a) that

G 2,2 (ρ) ≤ c • ρv ρ ∞ 0 (S 1 ) 3 r 3 e -1 2 πr 2 λ 2 (u 1:3 )|u 1 -u 2 | -1 1 r 2 λ2(u1:3)<vρ 1 u1 =u2 dσ(u 1:3 )dr.
Let φ 3 be the change of variables

φ 3 :[0, 2π) 3 -→ (S 1 ) 3 θ 1:3 -→ u 1:3 with u 1 = u(-θ 1 + θ 3 ), u 2 = u(θ 1 + θ 3 ) and u 3 = u(θ 2 + θ 3 ),
where u(θ) = (cos θ, sin θ). For all θ 1:3 ∈ (0, 2π) × [0, 2π) 2 , let us denote by A(θ 1:3 ) = λ 2 (u 1:3 ) with

u 1:3 = φ 3 (θ 1:3 ). Since |u 1 -u 2 | = 2| sin θ 1 |, we have G 2,2 (ρ) ≤ c • ρv ρ ∞ 0 (0,π/2)×[0,2π) 2 r 3 e -1 2 πr 2 A(θ 1:3 )| sin θ 1 | -1 1 r 2 A(θ1:3)<vρ dθ 1:3 dr.
Without loss of generality, we have assumed that θ 1 belongs to (0, π/2). Let > 3 5 be fixed. The previous inequality can be written as

G 2,2 (ρ) ≤ c • ρv ρ ∞ 0 (0,ρ -[×[0,2π) 2 r 3 e -1 2 πr 2 A(θ 1:3 )| sin θ 1 | -1 1 r 2 A(θ1:3)<vρ dθ 1:3 dr + c • ρv ρ ∞ 0 [ρ -,π/2)×[0,2π) 2 r 3 e -1 2 πr 2 A(θ 1:3 )| sin θ 1 | -1 1 r 2 A(θ1:3)<vρ dθ 1:3 dr = G (1) 2,2 (ρ) + G (2)
2,2 (ρ), (3.51) where

G (1)
2,2 (ρ) and G

(2)

2,2 (ρ) denote respectively the first and the second terms of the right-hand side. Let us note that A(θ 1:3 )| sin θ 1 | -1 is bounded since, according to (3.47), we have A(θ

1:3 ) = 1 2 • 2| sin θ 1 | • δ(u 3 , L(u 1:2 ))
, where u 1:3 = φ 3 (θ 1:3 ) and δ(u 3 , L(u 1:2 )) ≤ 2. Hence, the first integral of the right-hand side of (3.51) is less than

G (1) 2,2 (ρ) ≤ c • ρv ρ ∞ 0 (0,ρ -)×[0,2π) 2 r 3 e -1 2 πr 2 dθ 1:3 dr ≤ c • ρ 1-v ρ = O(ρ -1/5 ) (3.52)
since v ρ = c • ρ -3/5 and > 3 5 . Moreover, bounding A(θ 1:3 ) by r -2 v ρ in the second integral of (3.51), we have 1. To the best of our knowledge, there is no more accurate result on the Taylor expansion of K 1/6 (•) which could provide the rate of convergence of P (λ 2 (C) < v). Actually, the rate of convergence can be investigated by using the following expression of the density of λ 2 (C) due to Rathie [START_REF] Rathie | On the volume distribution of the typical Poisson-Delaunay cell[END_REF]:

G (2) 2,2 (ρ) ≤ c • ρv 2 ρ ∞ 0 [ρ -,π/2)×[0,2π) 2 re -1 2 πr 2 | sin θ 1 | -1 dθ 1:3 dr ≤ c • log ρ • ρv 2 ρ = O log ρ • ρ -1/5 (3.53) since π/2 ρ - 1 | sin θ1|
f (x) = ∞ k=0 c k,1 x 2/3+2k + ∞ k=0 c k,2 x 1+2k + ∞ k=0 c k,3 x 4/3+2k
for some constants c k,i , k ≥ 0, 1 ≤ i ≤ 3. It results from a Taylor expansion that P (λ

2 (C) < v) = c 0,1 • v 5/3 + O(v 2 ) for all v ≥ 0. Taking v = v ρ as in (3.42), we obtain G 1 (ρ) = |ρP (λ 2 (C) < v ρ ) -t 5/3 | = O ρ -1/5 so that P α 3/5 3 ρ 3/5 A P DT,min (ρ) ≥ t -e -t 5/3 = O log ρ • ρ -1/5 .
2. When d ≥ 3, we can show that the density of λ d (C) (see (2.5) in [START_REF] Rathie | On the volume distribution of the typical Poisson-Delaunay cell[END_REF])

) satisfies f (x) = c • x + o(x). This implies ρP λ d (C) < c • ρ -1/2 t -→ ρ→∞ t 2 .
Unfortunately, the same method as in the proof of Proposition 6 is not enough to show that G 2 (ρ) converges to 0.

Extreme Values of a Poisson-Voronoi tessellation

Let χ be a locally finite subset of R d . For all x ∈ χ, we denote by C χ (x) the Voronoi cell of nucleus x defined as

C χ (x) = {y ∈ R d : |x -y| ≤ |x -y|, x ∈ χ}.
For all x ∈ χ, we denote by N χ (x) the set of neighbors of x and by N χ (x) its cardinality, i.e.

N χ (x) = {x ∈ χ, C χ (x ) ∩ C χ (x) = ∅} and N χ (x) = #N χ (x).
We also consider the two quantities When χ = X is a Poisson point process (of intensity 1), the family m P V T = {C X (x), x ∈ X} is called the Poisson-Voronoi tessellation. Such a model is extensively used in many domains such as cellular biology [START_REF] Poupon | Voronoi and Voronoi-related tessellations in studies of protein structure and interaction[END_REF], astrophysics [START_REF] Ramella | Finding galaxy clusters using Voronoi tessellations[END_REF], telecommunications [START_REF] Baccelli | Stochastic Geometry and Wireless Networks Volume 2: Applications[END_REF] and ecology [START_REF] Roque | Introduction to Voronoi Diagrams with Applications to Robotics and Landscape Ecology[END_REF]. For a complete account, we refer to the books [START_REF] Møller | Lectures on random Voronoi tessellations[END_REF][START_REF] Okabe | Spatial tessellations: concepts and applications of Voronoi diagrams[END_REF][START_REF] Schneider | Stochastic and integral geometry. Probability and its Applications[END_REF] and the survey [START_REF] Calka | Tessellations. In New perspectives in stochastic geometry[END_REF].

D(C χ (x)) = max x ∈N χ(x)
As in Section 4, the window

W ρ = ρ 1/d [0, 1] d is partitioned into N ρ = ρ 2 log ρ 1/d d sub-cubes i ∈ V ρ .
The event A ρ is the same as in (3.2) and we can show that it satisfies Condition (FRC) for the Poisson-Voronoi tessellation with arguments very similar to the proof of Lemma 5. For each cell C = C X (x) ∈ m P V T , we take z(C X (x)) = x. A consequence of Slivnyak-Mecke theorem (see e.g. Theorem 3.3.5 in [START_REF] Schneider | Stochastic and integral geometry. Probability and its Applications[END_REF]) shows that the typical cell C satisfies the equality in distribution C D = C X∪{0} (0). Besides, the function G 2 (•) defined in (1.6) has an integral representation. Indeed, from Slivnyak-Mecke formula, it can be written as

G 2 (ρ) = ρ Cρ P f (C X∪{0,y} (0)) > v ρ , f (C X∪{0,y} (y)) > v ρ dy. ( 4.1)
Extremes of characteristic radii of Poisson-Voronoi tessellation are studied in [START_REF] Calka | Extreme values for characteristic radii of a Poisson-Voronoi tessellation[END_REF]. In this paper, we give the asymptotic behaviours of two new geometrical characteristics: Actually, the following proposition shows that the two random variables D d min,P V T (ρ) and F min,P V T (ρ) are of same order when ρ goes to infinity. Proposition 7. Let m P V T be a Poisson-Voronoi tessellation of intensity γ = 1. For all t ≥ 0, we have

P α 1/(d+1) d,4 ρ 1/(d+1) D d min,P V T (ρ) ≥ t -e -t d+1 = O ρ -1/(d+1) , (4.2a) P α 1/(d+1) d,5 ρ 1/(d+1) F min,P V T (ρ) ≥ t -e -t d+1 = O ρ -1/(d+1) , (4.2b)
where α d,4 and α d,5 are defined in (4.7) and (4.17) respectively.

Before proving Proposition 7, we need a practical lemma which is a new version of Lemma 3 in [START_REF] Calka | Extreme values for characteristic radii of a Poisson-Voronoi tessellation[END_REF] adapted to our framework. Lemma 10. Let v ≥ 0, y = 0 ∈ R d and χ ⊂ R d locally finite such that χ ∪ {0, y} is in general position, i.e. each subset of size n<d + 1 is affinely independent (see [START_REF] Zessin | Point processes in general position[END_REF]). Let us assume that each Voronoi cell associated with the set χ ∪ {0, y} is bounded and that N χ∪{0,y} (0) ⊂ B(0, v) and N χ∪{0,y} (y) ⊂ B(y, v).

(4.3)

Then # (χ ∩ (B(0, v) ∪ B(y, v))) ≥ d + 1.
Proof of Lemma 10. Let us define χ 0,y as the (finite) subset

χ 0,y = χ ∩ (B(0, v) ∪ B(y, v)) .
Thanks to (4.3), we have C χ∪{0,y} (0) = C χ0,y∪{0,y} (0) and C χ∪{0,y} (y) = C χ0,y∪{0,y} (y). In particular, this shows that the cells C χ0,y∪{0,y} (0) and C χ0,y∪{0,y} (y) are bounded. Hence 0 and y are in the convex hulls of χ 0,y ∪ {y} and χ 0,y ∪ {0} respectively (see Property V2, p. 58 in [START_REF] Okabe | Spatial tessellations: concepts and applications of Voronoi diagrams[END_REF]). This implies {0, y} ⊂ conv(χ 0,y ). Since χ ∪ {0, y} is in general position, this shows that conv(χ 0,y ) has a nonempty interior and consequently this proves Lemma 10.

We can now prove Proposition 7.

Proof of Proposition 7.

Proof of (4.2a). To find a function

v ρ = v ρ (t) such that G 1 (ρ) = |ρP (D(C) > v ρ )
-t| converges to 0, we have to approximate the tail of the distribution function of D(C). First, we notice that for all v > 0 we have

D (C χ (x)) < v ⇐⇒ N χ (x) ⊂ B(0, v), x ∈ χ (4.4)
for any locally finite subset χ of R d . This fact applied to χ = X ∪ {0} shows that

P (D(C) < v) = ∞ k=d+1 P N X∪{0} (0) ⊂ B(0, v), N X∪{0} (0) = k (4.5)
since C = C X∪{0} (0). An integral representation of the right-hand side is given by (see Proposition 1 in [START_REF] Calka | Precise formulae for the distributions of the principal geometric characteristics of the typical cells of a two-dimensional Poisson-Voronoi tessellation and a Poisson line process[END_REF])

P N X∪{0} (0) ⊂ B(0, v), N X∪{0} (0) = k = 1 k! B(0,v) k e -λ d (F (C {x 1:k }∪{0} (0))) 1 F k (x 1:k )dx 1:k ,
where

F k = x 1:k = (x 1 , . . . , x k ) ∈ (R d ) k , C {x 1:k }∪{0} (0) is a convex polytope with k facets .
We recall that {x 1:k } ∪ {0} = {x 1 , x 2 , . . . , x k , 0}. Taking the change of variables x i = vx i , we obtain for all k ≥ d + 1 d+1) when v goes to 0, where

P N X∪{0} (0) ⊂ B(0, v), N X∪{0} (0) = k = v dk • 1 k! B(0,1) k e -v d λ d (F (C {x 1:k }∪{0} (0))) 1 F k (x 1:k )dx 1:k . (4.6) If k = d + 1, the previous probability is equivalent to α d,4 • v d(
α d,4 = 1 (d + 1)! B(0,1) d+1 1 F d+1 (x 1:d+1 )dx 1:d+1 . ( 4.7)
If k ≥ d + 2, the right-hand side of (4.6) is less than

κ k d k! v dk thanks to 1 F k ≤ 1 and e -λ d (F (C {x 1:k }∪{0} (0))) ≤ 1. It follows from (4.5) that P (D(C) < v) -α d,4 • v d(d+1) ≤ ∞ k=d+2 κ k d k! v dk = O(v d(d+2) ). (4.8)
Now, we can choose a suitable function v ρ . Indeed, let t ≥ 0 be fixed and let us denote by

v ρ = v ρ (t) = α -1 d,4 ρ -1 1/d(d+1) t 1/d . ( 4.9)
According to (4.8), we have To guarantee the independence of the events considered in (4.11) for each cells which are distant enough, we write

G 1 (ρ) = |ρP (D(C) < v ρ ) -t d+1 | = O ρ -1/(d+1) . ( 4
G 2 (ρ) = ρ Cρ∩B(0,2vρ) c P N X∪{0,y} (0) ⊂ B(0, v ρ ), N X∪{0,y} (y) ⊂ B(y, v ρ ) dy + ρ Cρ∩B(0,2vρ) P N X∪{0,y} (0) ⊂ B(0, v ρ ), N X∪{0,y} (y) ⊂ B(y, v ρ ) dy. (4.12)
For the first integral, when y ∈ C ρ ∩ B(0, 2v ρ ) c , the balls B(0, v ρ ) and B(y, v ρ ) are disjoint. Because X is a Poisson point process and because y ∈ B(0, 2v ρ ), the first integrand of (4.12) can be written as the product P N X∪{0} (0) ⊂ B(0, v ρ ) × P N X∪{y} (y) ⊂ B(y, v ρ ) . Hence, according to (4.4) and (4.10) we obtain that

P N X∪{0,y} (0) ⊂ B(0, v ρ ), N X∪{0,y} (y) ⊂ B(y, v ρ ) = P (D(C) < v ρ ) 2 ≤ c • ρ -2 , ( 4.13)
where y ∈ R d \B(0, 2v ρ ) and where c is a constant which does not depend on y.

For the second integral of (4.12), we apply Lemma 10 to χ = X. This gives

P N X∪{0,y} (0) ⊂ B(0, v ρ ), N X∪{0,y} (y) ⊂ B(y, v ρ ) ≤ P (#(X ∩ (B(0, v ρ ) ∪ B(y, v ρ )) ≥ d + 1)) , ( 4.14)
where y ∈ B(0, 2v ρ ). Since #(X ∩ B) is Poisson distributed of mean λ d (B) for each Borel subset B ⊂ R d , we obtain for ρ large enough that

P (#(X ∩ (B(0, v ρ ) ∪ B(y, v ρ )) ≥ d + 1)) = ∞ k=d+1 1 k! (λ d (B(0, v ρ ) ∪ B(y, v ρ ))) k e -λ d (B(0,vρ)∪B(y,vρ)) ≤ c • v d(d+1) ρ = c • ρ -1
according to (4.9) and to the inequalities e -λ d (B(0,vρ)∪B(y,vρ)) ≤ 1 and

λ d (B(0, v ρ ) ∪ B(y, v ρ )) ≤ 2 • κ d v d ρ
, with y ∈ B(0, 2v ρ ). This together with (4.12), (4.13) and (4.14) shows that

G 2 (ρ) ≤ c • ρ -1 λ d (C ρ ∩ (R d \B(0, 2v ρ ))) + c • λ d (C ρ ∩ B(0, 2v ρ )). Since λ d (C ρ ∩ B(0, 2v ρ ) c ) ≤ λ d (C ρ ) ≤ c • log ρ and λ d (C ρ ∩ B(0, 2v ρ )) ≤ λ d (B(0, 2v ρ )) = c • ρ -1/(d+1) , we deduce from the previous inequality that G 2 (ρ) ≤ c • log ρ × ρ -1 + c • ρ -1/(d+1) = O ρ -1/(d+1) . ( 4

.15)

We now derive directly (4.2a) from (4.10), (4.15) and Theorem 1.

Proof of (4.2b). This will be sketched since it is analogous to the proof of (4.2a). First, we investigate the tail of the distribution function of λ d (F(C)). In [START_REF] Zuyev | Estimates for distributions of the Voronoi polygon's geometric characteristics[END_REF], Zuyev shows that, conditional on N X∪{0} = k, the volume of F(C) is Gamma distributed with parameters (k, 1), i.e.

P (λ

d (F(C)) < v) = ∞ k=d+1 1 (k -1)! v 0 x k-1 e -x dx • p(k), ( 4
|P (λ d (F(C)) < v) -α d,5 • v d+1 | = O(v d+2 ).
Hence, for all fixed t ≥ 0, taking

v ρ = v ρ (t) = α -1 d,5 ρ -1 1/(d+1) t (4.18)
we obtain

G 1 (ρ) = |ρP (λ d (F(C)) < v ρ ) -t d+1 | = O(ρ -1/(d+1) ). ( 4

.19)

To get an upper bound for G 2 (ρ), we note that for each χ ⊂ R d locally finite and x ∈ χ, we have

κ d 2 d • (D(C χ (x))) d ≤ λ d (F(C χ (x))) .
Applying the previous inequality to χ = X ∪ {0, y} and x = 0, y, we deduce from (4.1) that

G 2 (ρ) = ρ Cρ P λ d (F(C X∪{0,y} (0))) < v ρ , λ d (F(C X∪{0,y} (y))) < v ρ dy ≤ ρ Cρ P D(C X∪{0,y} (0)) < v ρ , D(C X∪{0,y} (y)) < v ρ dy (4.20) with v ρ = v ρ (t) = 2κ 1/d d • v 1/d ρ = (2 d(d+1) κ d+1 d α -1 d,5 ρ -1
) 1/d(d+1) t 1/d according to (4.18). Let us notice that there exists a constant c such that, for any t ≥ 0 and ρ ≥ 0, we have v ρ (t) = c • v ρ (t), where v ρ = v ρ (t) is defined in (4.9). Writing the right-hand side of (4.20) in the same spirit as in (4.11) and proceeding along the same lines as in the proof of (4.2a), we show that G 2 (ρ) is of order ρ -1/(d+1) . This together with (4.19) shows (4.2b).

A similar method could be used to re-find the asymptotic behaviour of the minimum of circumradii given in [START_REF] Calka | Extreme values for characteristic radii of a Poisson-Voronoi tessellation[END_REF].

The maximum of inradii of a Gauss-Poisson Voronoi tessellation

As an example of non-Poisson point process, a Gauss-Poisson process is analyzed. Introduced by Newman [START_REF] Newman | A new family of point processes which are characterized by their second moment properties[END_REF] and investigated by Milne and Westcott, such process has a potential application in statistical mechanics (see [START_REF] Newman | A new family of point processes which are characterized by their second moment properties[END_REF], p. 350) and could be used as a model for molecular motion (see [START_REF] Milne | Further results for Gauss-Poisson Process[END_REF] p. 169). In the sense of [START_REF] Stoyan | Stochastic Geometry and its applications[END_REF] p. 161, a stationary planar Gauss-Poisson process X is a (simple) point process which can be defined as follows. Let X a be a Poisson point process of intensity γ a in R 2 . Every point x a ∈ X a is replaced by a cluster of points Ξ(x a ) = x a + Ξ 0 (x a ), where the sets of points Ξ 0 (x a ) are chosen independently and with identical distribution, i.e. X = xa∈Xa (x a + Ξ 0 (x a )).

(5.1) For all t ∈ R, we define v ρ = v ρ (t) so that P (v ρ ) = log ρ + t, i.e. Using the fact that ρP 0 (#X 0 ∩ B(0, 2v ρ ) = 1) = e -t-R(vρ) , we deduce that G 1 (ρ) = |ρP 0 (#X 0 ∩ B(0, 2v) = 1) -e -t | ≤ e -t R(v ρ ) = O (log ρ) -1/2 .

v ρ = v ρ (t) = 2γ a • p 2 + 4γ
(5.6)

To check Condition (LCC), we first provide an integral representation for G 2 (ρ) with respect to Palm measures. Then we give some upper bound for the integrand. To do it, we denote by F lf the space of locally finite subsets of R 2 . We deduce from Campbell's theorem (see Theorem 3.3.3. in [START_REF] Schneider | Stochastic and integral geometry. Probability and its Applications[END_REF]) that since the integrand of the right-hand side of the second line is translation invariant (in distribution) and N ρ λ 2 (C ρ ) = c • ρ. According to Formula (5.3.2) in [START_REF] Stoyan | Stochastic Geometry and its applications[END_REF], we have P 0 = P X * c 0 , where P X is the distribution of X and c 0 is the Palm measure of the cluster distribution Ξ 0 that is concentrated on the space F lf,2 of subsets of 0, 1 or 2 points in R 2 . Hence When |y| > 2v ρ , we have y ∈ ξ for ρ large enough since c 0 a.s. ξ is bounded. Moreover, P X a.s. φ ∩ ξ ∩ (B(0, 2v ρ ) ∪ B(y, 2v ρ )) is empty. Consequently, calculating the integral with respect to c 0 and proceeding as previously, we deduce from Campbell's theorem and from the relation P 0 = P X * c 0 that

G 2 (ρ) = N ρ E   (x,y) = ∈(X∩Cρ)
G 2 (ρ) ≤ c • ρ F lf y∈φ∩Cρ 1 #φ∩(B(0,2vρ)∪B(y,2vρ))=1 1 |y|>2vρ dP X (φ) = c • ρ Cρ F lf F lf,2
1 #((ξ∪φ)+y)∩(B(0,2vρ)∪B(y,2vρ))=1 1 |y|>2vρ dc 0 (ξ)dP X (φ)dy.

Since P X a.s. φ ∩ ξ ∩ (B(0, 2v ρ ) ∪ B(y, 2v ρ )) is empty, we deduce after integration over F lf × F lf,2 with respect to P X ⊗ c 0 that G 2 (ρ) ≤ c • ρ Cρ P (X ∩ (B(0, 2v ρ ) ∪ B(y, 2v ρ )) = ∅) 1 |y|>2vρ dy.

(5.7)

We provide below a suitable upper bound for the integrand. Let |y| > 2v ρ be fixed. First, we note that X ∩ (B(0, 2v ρ ) ∪ B(y, 2v ρ )) = ∅ if and only if (x + Ξ 0 (x)) ∩ (B(0, 2v ρ ) ∪ B(y, 2v ρ )) = ∅ for all x ∈ X a . From Theorem 3.2.4. of [START_REF] Schneider | Stochastic and integral geometry. Probability and its Applications[END_REF], Fubini's theorem and the fact that Ξ 0 is symmetric with respect to the origin, we get where α = p1 2(p1+p2) . Since p 1 = 0, we have α > 0 so that G 2 (ρ) converges to 0. Proposition 8 is now a direct consequence of (5.6), (5.11) and Theorem 1.

According to Proposition 8 and (5.5), the order of E [r max,GP V T (ρ)] is (4γ a π(p 1 + p 2 ))

-1/2 • (log ρ) • (log ρ) 1/2 since we have assumed that (p 1 + 2p 2 )γ a = 1. Let us remark that the larger p 2 is, the larger the order is. This can be explained by the following heuristic fact: the nucleus x ∈ X of the Voronoi cell which maximizes the inradius belongs to a cluster of size 1, i.e. x ∈ Ξ(x a ), where #Ξ(x a ) = 1 for some x a ∈ X a . Hence if p 2 is large, the mean number of clusters of size 1 is small so that the inradii associated with the clusters of size 1 are large.

When p 1 = 0, we obtain a degenerate case since r max,GP V T (ρ) is constant. Indeed, with high probability, the maximum of inradius is made between two points which belong to the same cluster so that r max,GP V T (ρ) = 1 2 . When p 0 = p 2 = 0 and p 1 = 1, the random variable r max,GP V T (ρ) is the maximum of inradii of a Poisson-Voronoi tessellation r max,P V T (ρ). In that case, the order is

v ρ = v ρ (t) = (4π) -1/2 • (log ρ + t) 1/2 .
The order of r max,P V T (ρ) has already been investigated in [START_REF] Calka | Extreme values for characteristic radii of a Poisson-Voronoi tessellation[END_REF]. Nevertheless, Proposition 8 is more precise since it provides a rate of convergence. Actually, this rate could be improved. Indeed, since p 0 = p 2 = 0 and p 1 = 1 we have R(v ρ ) = 0 according to (5.2) and (5.4) and consequently we get G 1 (ρ) = 0 according to the inequality in (5.6). Moreover, the term α which appears in (5.11) equals 1/2. Hence, according to (5.11), we obtain the more precise result:

P r max,P V T (ρ) ≤ (4π) -1/2 • (log ρ + t)

1/2 = O log ρ • ρ -1/2 .
Finally, let us mention that a Gauss-Poisson process belongs to the class of the so-called Neyman-Scott processes (see section 5.3 in [START_REF] Stoyan | Stochastic Geometry and its applications[END_REF]). We do not investigate general Neyman-Scott processes since we cannot provide a Taylor expansion of P (r(C) > r) in general.

  and N ρ -→ ρ→∞ ∞. These sub-cubes are indexed by the set of i = (i 1 , . . . , i d ) ∈ 1, N 1/d ρ d

P 1 :

 1 f (ru 1:k , y k+1:d+1 ) > v ρ , R(ru 1:k , y k+1:d+1 ) ≤ r and z(ru 1:k , y k+1:d+1 ) ∈ C ρ , (3.6a) P 2 : y j ∈ B(ru 1:d+1 ) and ru j ∈ B(ru 1:k , y k+1:d+1 ) for all k + 1 ≤ j ≤ d + 1.(3.6b)The first property concerns the cell ∆ 2 which has the smallest circumradius whereas the second property means that the two simplices are Delaunay cells. Moreover, we introduce the setE k,r,u 1:d+1 = {y k+1:d+1 ∈ (R d ) d+1-k satisfying P 1 and P 2 }.(3.7)Finally, in the same spirit as in (3.5), we consider the volume of the union of the two circumballs, i.e. λ (∪) d (r, u 1:k , y k+1:d+1 ) = λ d (B(0, r) ∪ B(ru 1:k , y k+1:d+1 )) . (3.8)

  r, u 1:d+1 , y k+1:d+1 )dy k+1:d+1 dσ(u 1:d+1 )dr (3.10) andg 2,k (ρ, r, u 1:d+1 , y k+1:d+1 ) = r d 2 -1 e -β d λ (∪) d (r,u 1:k ,y k+1:d+1 ) λ d (u 1:d+1 )1 f (ru 1:d+1 )>vρ 1 E k,r,u 1:d+1 (y k+1:d+1 ).(3.11)

. 27 )of Lemma 7 .

 277 Proof Let {x 1:3 } and {x 1:3 } be fixed.If the interior of B(x 1:3 ) ∩ B(x 1:3 ) is empty, we have λ 2 (B(x 1:3 ) ∪ B(x 1:3 )) = λ 2 (B(x 1:3 )) + λ 2 (B(x 1:3 )) ≥ πR 2 + λ 2 (x 1:3 ).(3.28)

Figure 1 :

 1 Figure 1: (a). A union of two disks. (b). The triangle which maximizes the area.

Proposition 6 . 3 / 5 3

 635 Let m P DT be a Poisson-Delaunay tessellation of intensity γ = 1 in R2 . Then for all t ≥ 0 P α ρ 3/5 A min,P DT (ρ) ≥ t -→ ρ→∞ e -t 5/3 , (

  dθ 1 is of order log ρ. From (3.50), (3.51), (3.52) and (3.53), we deduce that G 2 (ρ) = O log ρ • ρ -1/5 . Proposition 6 is now a direct consequence of (3.43) and Theorem 1.

  |x -x |, and F(C χ (x)) = y∈Cχ(x) B(y, |y -x|) which are the distance to the farthest neighbor and the so-called Voronoi flower of nucleus x ∈ χ. A Voronoi tessellation corresponds to the dual graph of Delaunay tessellation in the following way: there exists an edge between two points x, x ∈ χ in the Delaunay graph if and only if they are Voronoi neighbors, i.e. C χ (x) ∩ C χ (x ) = ∅.

D

  min,P V T (ρ) = min x∈X∩Wρ D(C X (x)) and F min,P V T (ρ) = min x∈X∩Wρ λ d (F(C X (x))). Obviously, 2 -d κ d D d min,P V T (ρ) ≤ κ d min x∈X∩Wρ R(C X (x)) d ≤ F min,P V T (ρ).

. 10 )

 10 Let us give now an upper bound for the function G 2 (ρ) defined in(1.6). According to (4.1) and (4.4), we obtainG 2 (ρ) = ρ Cρ P D(C X∪{0,y} (0)) < v ρ , D(C X∪{0,y} (y)) < v ρ dy = ρ Cρ P N X∪{0,y} (0) ⊂ B(0, v ρ ), N X∪{0,y} (y) ⊂ B(y, v ρ ) dy.

2

 2

1

  #X∩B(x,2vρ)=1 1 #X∩B(y,2vρ)=1   = N ρ Cρ F lf y∈η∩Cρ 1 #(η+x)∩B(x,2vρ)=1 1 #(η+x)∩B(y,2vρ)=1 dP 0 (η)dx = c • ρ F lf y∈η∩Cρ 1 #η∩B(0,2vρ)=1 1 #η∩B(y,2vρ)=1 dP 0 (η)

G 2

 2 (ρ) = c • ρ F lf F lf,2 y∈(φ∪ξ)∩Cρ1 #(φ∪ξ)∩(B(0,2vρ)∪B(y,2vρ))=2 1 |y|>2vρ dc 0 (ξ)dP X (φ).

PE= p 1 + 2p 2 p 1 P 2 ρ( 5 . 10 )

 12510 (X ∩ (B(0, 2v ρ ) ∪ B(y, 2v ρ )) = ∅) = e -γa R 2 P((x+Ξ0(x))∩(B(0,2vρ)∪B(y,2vρ)) =∅)dx = e -γaE[λ2(Ξ0+(B(0,2vρ)∪B(y,2vρ)))] .(5.8)We give below a suitable lower bound for the term appearing in the exponential. Since |y| > 2v ρ , we haveE [λ 2 (Ξ 0 + (B(0, 2v ρ ) ∪ B(y, 2v ρ )))|#Ξ 0 = 1] = λ 2 (B(0, 2v ρ ) ∪ B(y, 2v ρ )) [λ 2 (Ξ 0 + (B(0, 2v ρ ) ∪ B(y, 2v ρ )))|#Ξ 0 = 2] ≥ E [λ 2 (Ξ 0 + B(0, 2v ρ ))] ≥ 8πv 2 ρ -a(2v ρ ), where a(•) is defined in(5.3). Since Ξ 0 is reduced to 0, 1 or 2 points with probability p 0 , p 1 and p 2 respectively, we deduce from (5.8) thatP (X ∩ (B(0, 2v ρ ) ∪ B(y, 2v ρ )) = ∅) ≤ e -γa( 3 2 p1•4πv 2 ρ +p2(8πv 2 ρ -a(2vρ))) 0 (#B(0, 2v ρ ) ∩ X 0 = 1) • e -2γap1πvfor ρ large enough according to (5.2). Integrating over C ρ , we deduce from (5.6), (5.7), (5.10) and from the inequalityλ 2 (C ρ ) ≤ c • log ρ, that G 2 (ρ) ≤ c • log ρ • e -2γap1πv 2 ρ = O log ρ • ρ -α ,(5.11)

  Then the family of point processes Φ ρ converges in distribution to the Poisson point process Φ, i.e. for any Borel subsets B 1 , . . . , B k

  X∪{0} (0) = k . When k = d + 1, the Taylor expansion e -x = 1 + O(x) shows that the first term of the series in (4.16) equals α d,5 v d+1 + O(v d+2 ), where

	α d,5 =	p(d + 1) (d + 1)!	.	(4.17)
	If k ≥ d + 2, the term of the series in (4.16) is less than 1 d! • v d+2 • p(k) thanks to the inequality e -x ≤ 1. According
	to (4.16), we get

.16)

where p(k) = P N

  2 a • p 2 2 + 4γ a π(p 1 + p 2 ) log

	4γ a π(p 1 + p 2 )	p1 p1+2p2 + log ρ + t	1/2	.	(5.5)

  1/2 = p 1 + 2p 2 4π(p 1 + p 2 )

	1/2

which is the number of exceedance pairs.

Acknowledgements. I would like to thank my advisor P. Calka for helpful discussions and suggestions. This work was partially supported by the French ANR grant PRESAGE (ANR-11-BS02-003) and the French research group GeoSto (CNRS-GDR3477).

For all x a ∈ X a , the cluster Ξ 0 (x a ) equals in distribution Ξ 0 which is defined in the following way: Ξ 0 has an isotropic distribution and is composed of zero, one or two points with probability p 0 = 1, p 1 and p 2 = 1 -(p 0 + p 1 ). If Ξ 0 contains only one point then that point is the origin 0. If Ξ 0 is composed of two points then these are separated by a unit distance and have midpoint 0. The intensity of X is given by γ X = (p 1 + 2p 2 ) • γ a . In this subsection, we investigate the maximum of inradii of a Gauss-Poisson Voronoi tessellation m GP V T , i.e.

To apply Theorem 1, we subdivide

where v ρ (t) is given in (5.5).

Proof of Proposition 8. We notice that for all x ∈ X and v ≥ 0, the inradius r(C X (x)) is greater than v if and only if #X ∩ B(x, 2v) = 1. Consequently

where C is the typical cell of the Voronoi tessellation induced by X. In the above equality, P 0 is the Palm measure of X in the sense of (3.6) of [START_REF] Schneider | Stochastic and integral geometry. Probability and its Applications[END_REF] and X 0 is P 0 distributed. The planar Gauss-Poisson process is one of the rare non-Poisson processes for which the right-hand side can be made fully explicit. It is given for each v ≥ 0 by (see p.

161 in [START_REF] Stoyan | Stochastic Geometry and its applications[END_REF]):

and

and equals zero otherwise. The function a(2v) is the area of the intersection of two disks of radius 2v and centers separated by unit distance. A Taylor expansion of the right-hand side of (5.2) shows that

where

as v goes to infinity. In the previous line, φ(v) = o(ψ(v)) means that φ(v)/ψ(v) -→ v→∞ 0.