Large Poisson-Voronoi Cells and Crofton Cells

Large Poisson-Voronoi Cells and Crofton Cells

Daniel Hug^∗ Matthias Reitzner^†

Rolf Schneider^‡

Abstract. It is proved that the shape of the typical cell of a stationary Poisson-Voronoi tessellation in Euclidean space, under the condition that the volume of the typical cell is large, must be close to spherical shape, with high probability. The same holds if the volume is replaced by the surface area or other suitable functionals. Similar results are established for the zero cell of a stationary and isotropic Poisson hyperplane tessellation.

Keywords: Poisson-Voronoi tessellation; typical cell; Poisson hyperplane tessella- tion; Crofton cell; asymptotic shape; D.G. Kendall’s conjecture; intrinsic volume

AMS 2000 Subject Classification: Primary 60D05

Secondary 52A22, 52A40

1 Introduction

In this paper, we continue a line of research that began with a problem of D.G. Kendall (see the foreword to the first edition of [17]). He conjectured that the shape of the zero cell (or Crofton cell) of a stationary and isotropic Poisson line process in the plane, given that the area of the cell tends to infinity, must become circular. Contributions to Kendall’s question are due to Miles [11] and Goldman [3], and the conjecture was proved by Kovalenko [8], [10]. In [7], Kovalenko’s result was extended to higher dimensions and to stationary, but not necessarily isotropic Poisson hyperplane processes. It was also strengthened, by estimating the probability of large deviations from spherical shape, given that the volume of the zero cell lies in a prescribed interval. In the present paper, we prove an analogous result for the typical cell of a stationary Poisson-Voronoi tessellation (mosaic) of d-dimensional space. Thus we extend and strengthen a result of Kovalenko [9] in the planar case. We further extend this result by considering, in addition to the volume functional, also the kth intrinsic volume, k= 1, . . . , d−1. This includes cells of large surface area or of large mean width. The result from [7] on Crofton cells of stationary Poisson hyperplane processes with large volume is also extended to thekth intrinsic volume, but only fork≥2 and under the additional assumption of isotropy. For both types of random polytopes, Poisson-Voronoi cells and isotropic Crofton cells, we can also replace (somewhat easier) the condition of large volume by the condition of

∗Postal address: Mathematisches Institut, Albert-Ludwigs-Universit¨at, Eckerstr. 1, D-79104 Freiburg, Germany. Email address: daniel.hug@math.uni-freiburg.de

†Postal address: Institut f¨ur Analysis und Technische Mathematik, Technische Universit¨at Wien, Wiedner Hauptstrasse 8 - 10, A-1040 Wien, Austria. Email address: mreitzne@mail.zserv.tuwien.ac.at

‡Postal address: Mathematisches Institut, Albert-Ludwigs-Universit¨at, Eckerstr. 1, D-79104 Freiburg, Germany. Email address: rolf.schneider@math.uni-freiburg.de

large inradius. This is suggested by considerations of Miles [11] on Crofton cells in the plane and by the work of Calka [1] on planar Poisson-Voronoi tessellations. Finally, we mention here that cells of large volume in Poisson-Delaunay tessellations were treated in [6]; such cells tend to be regular simplices.

Let A be a locally finite point set in Euclidean space R^d (with scalar product h·,·i and normk · k), where d≥2. For x∈A, theVoronoi cellof xwith respect toA is defined by

C(x, A) :={y∈R^d:ky−xk ≤ ky−ak for all a∈A}.

Let ˜X be a stationary Poisson point process of intensity λ > 0 in R^d. (In treating simple point processes, we conveniently identify a simple counting measure with its support.) Then

X :={C(x,X) :˜ x∈X}˜

is the Poisson-Voronoi tessellation derived from ˜X. Let Z denote the typical cell of X (we recall its definition in Section 2).

For a convex body (non-empty, compact, convex set) K ⊂R^d, we denote the volume by v_d(K). The (conveniently renormalized)intrinsic volumesv0(K), . . . , vd−1(K) can be defined by means of the Steiner formula

vd(K+B^d) =

d

X

k=0

^d−k d

k

vk(K), ≥0.

Here B^d := {x ∈ R^d :kxk ≤ 1} is the unit ball. Equivalently, v_i(K) is the mixed volume V(K, . . . , K, B^d, . . . , B^d), whereKappearsitimes andB^dappearsd−itimes. The functional W_j =vd−j is known as thejthquermassintegral. In particular,dvd−1 is the surface area, and (2/κ_d)v₁ is the mean width; hereκ_d:=v_d(B^d). More information is found in [15].

LetK ⊂R^d be a compact set witho∈K and containing more than one point. In order to measure the deviation ofK from a ball with centreo, we define

ϑ(K) := R_o−ρ_o Ro+ρo

,

whereR_o is the radius of the smallest ball with centre ocontaining K and ρ_o is the radius (possibly zero) of the largest ball with centreocontained inK.

By P we denote the underlying probability, and P(· | ·) is a conditional probability.

Theorem 1. Let X denote the Poisson-Voronoi tessellation derived from a stationary Poisson point process with intensityλ >0 in R^d; let Z be its typical cell. Let k∈ {1, . . . , d}. There is a positive constant c₀ depending only on the dimension d such that the following is true. If ∈(0,1)and I = [a, b) is any interval (possibly b=∞) with a^d/kλ≥σ₀, for some constant σ0>0, then

P(ϑ(Z)≥|v_k(Z)∈I)≤cexpn

−c₀^(d+3)/2a^d/kλo , where c is a constant depending only ond, and σ0.

In particular,

a→∞lim P(ϑ(Z)≥|v_k(Z)≥a) = 0 for every >0, (1)

but Theorem 1 provides much stronger information. The relation (1) can equivalently be formulated as follows (see Section 2 for further explanation).

Corollary. The conditional law for the shape ofZ, given a lower bound forvk(Z), converges weakly, as that lower bound tends to infinity, to the law concentrated at the shape of a ball.

Theorem 1 will be proved in Section 6, after preliminary explanations in Section 2 and preparations in Sections 3 to 5.

In [7], a similar result was obtained for the volume of the zero cell (also called Crofton cell) of the tessellation generated by a stationary Poisson hyperplane process. We will indicate in Section 7 how, under the additional assumption of isotropy, this result can be extended to thekth intrinsic volume, k= 2, . . . , d. As in [7], we measure the deviation of the shape of a convex bodyK ⊂R^d with interior points from spherical shape byr_B^d, which we abbreviate byr_d, thus

rd(K) := min{s/r−1 :rB^d+z⊂K⊂sB^d+z,z∈R^d, r, s >0}.

Theorem 2. Let Zo be the zero cell of the tessellation induced by a stationary isotropic Poisson hyperplane process in R^d with intensity λ > 0. Let k ∈ {2, . . . , d}. There is a positive constant c₀ depending only on the dimension d such that the following is true. If ∈(0,1)and I = [a, b) is any interval (possibly b=∞) with a^1/kλ≥σ0, for some constant σ₀>0, then

P(rd(Zo)≥|vk(Zo)∈I)≤cexp n

−c₀^(d+3)/2a^1/kλ o

, where c is a constant depending only ond, and σ0.

The case of the volume is included here fork=d. We remark that in this case the inequality of Theorem 2 is sharper (in its dependence on) than Theorem 1 of [7], specialized to the isotropic case. The reason for this improvement lies in the fact that in the isotropic case sharper stability estimates from convex geometry are available.

Unfortunately, our method of proof does not permit us to treat the case k= 1, which in the plane is the case of the perimeter, already studied by Miles [11] in his heuristic approach.

In addition to ρ_o(K) defined above, we denote by ρ(K) the radius of the largest ball contained in the convex bodyK.

Theorem 3. Let Z and Z_o be defined as in Theorems 1 and 2, respectively. There is a positive constant c0 depending only on the dimension d such that the following is true. If ∈ (0,1) and I = [a, b) is any interval (possibly b = ∞) with a^dλ ≥ σ₀ in the case of Z, respectivelyaλ≥σ₀ in the case of Z_o, with some constantσ₀>0, then

P(ϑ(Z)≥|ρ_o(Z)∈I)≤cexpn

−c₀^(d+1)/2a^dλo and

P(rd(Zo)≥|ρ(Zo)∈I)≤cexp n

−c₀^(d+1)/2aλ o

, where c is a constant depending only ond, and σ0.

The proof will be sketched in Section 8.

A few words about the choices of the shape parameters ϑ(K) and r_d(K) seem in order.

For the formulation of our estimates, we want a simple, similarity invariant measure for the deviation of the shape of a convex body from spherical shape. Such a measure appears implicitly in relation (2) of Miles [11], and explicitly in the paper of Kovalenko [10]. The numberrd(K) used above is the extension of the latter to higher dimensions. For an interior pointz of the compact convex setK, let Rz be the radius of the smallest ball with centrez containingK, andρz the radius of the largest ball with centrezcontained inK. Thenr_d(K) is the minimum, over allz in the interior of K, of the quotient (Rz−ρz)/ρz. In the case of the typical cell of a Poisson-Voronoi tessellation, for which the nucleus ois a distinguished point, we get a sharper result if we use instead the deviation measure (Ro−ρo)/ρo, or, which for bodies close to a ball with centreois essentially equivalent, (Ro−ρo)/(Ro+ρo) =ϑ(K).

Here we have chosen the latter, since for this the crucial stability estimate (13) below takes a simple form. Finally, we mention that Goldman [3], in the planar case, considers the first eigenvalue of the Laplacian for the Dirichlet problem and finds the same asymptotic behaviour for large Crofton cells and their incircles, but this does apparently not lead to explicit geometric estimates for the deviation from circular shape.

2 The typical cell of a Poisson-Voronoi tessellation

In general, the notion of the typical cell of a stationary random tessellation requires the choice of a centroid function (e.g., see Møller [12, Section 3.2]), but for Voronoi cells there is a canonical choice, the nucleus. Let X be the Poisson-Voronoi tessellation generated by a stationary Poisson point process ˜X. Let K^d_o denote the space of convex bodies K in R^d with o ∈ K, equipped with the Hausdorff metric and corresponding Borel structure. The distributionQof the typical cell ofX can be defined by

Q(A) = 1 λEX

x∈X˜

1A(C(x,X)˜ −x)1_B(x)

(E denotes mathematical expectation) for Borel setsA ⊂ K^d_o, where B ⊂R^d is an arbitrary Borel set with λ_d(B) = 1; here λ_d denotes Lebesgue measure in R^d. The distribution Q is commonly interpreted as the conditional distribution of C(o,X) given that˜ o ∈ X. An˜ intuitive interpretation follows from the fact that stationary Poisson-Voronoi tessellations are mixing ([16, Satz 6.4.1]) and hence ergodic. This implies that

Q(A) = lim

r→∞

card{x∈X˜∩rB^d:C(x,X)˜ −x∈ A}

card ( ˜X∩rB^d) holds with probability one.

Since ˜Xis a stationary Poisson process, it follows from Slivnyak’s theorem that the typical cell of the Poisson-Voronoi tessellationX is equal in distribution to the random polytope

Z =C(o,X˜ ∪ {o})

(see [12, Remark 4.1.1]; see also [13]). Hence, we can considerZ as the typical cell of X, and for this we obtain a convenient representation. Forx∈R^d, we define

H(x) := {y∈R^d:hy,xi=kxk²/2}, H⁻(x) := {y∈R^d:hy,xi ≤ kxk²/2},

so thatH(x) is the mid hyperplane ofoand x. Then Z = \

x∈X˜

H⁻(x),

thusZ is the zero cell of the tessellation induced by the Poisson hyperplane process Y :={H(x) :x∈X}.˜

The intensity measure EY(·) of this process can be represented as follows (recall that Y denotes a random simple counting measure on the space of hyperplanes as well as its support).

For a Borel setAin the space of hyperplanes, we have

EY(A) = E card (A ∩Y) = E card{x∈X˜ :H(x)∈ A}

= λ·λ_d({x∈R^d:H(x)∈ A}).

Writing

H(u, t) :={y∈R^d:hy,ui=t}, H⁻(u, t) :={y∈R^d:hy,ui ≤t}

foru∈S^d−1 :={x∈R^d:kxk= 1} and t∈R, and introducing polar coordinates, we get EY(·) = 2^dλ

Z

S^d−1

Z ∞ 0

1{H(u, t)∈ ·}t^d−1dt σ(du), (2) whereσ denotes spherical Lebesgue measure on the unit sphere S^d−1.

In particular, for K ∈ K^d_o let H_K be the set of all hyperplanes H ⊂R^d withH∩K6=∅.

Then (2) gives

EY(H_K) = 2^dλU(K) (3)

with

U(K) := 1 d

Z

S^d−1

h(K,u)^dσ(du), (4)

whereh(K,·) is the support function ofK. Writing

Φ(K) :={y∈R^d:H(2y)∩K6=∅},

we have U(K) = λ_d(Φ(K)). The star body Φ(K) is the union of all closed balls having a diameter segment [o,x] withx∈K.

The relation (3), together with the Poisson property of the hyperplane processY, implies that

P(Y(H_K) =n) = [2^dU(K)λ]ⁿ

n! exp{−2^dU(K)λ} (5)

forK∈ K^d_o and n∈N0.

It is now clear that we are in a similar situation as in [7]. There, the zero cell of a stationary (not necessarily isotropic) Poisson hyperplane process, with intensity measure given by [7, (2)], was studied. This process is now replaced by the isotropic, non-stationary Poisson hyperplane processY, with intensity measure given by (2). The functionalU(K), defined by (4), will play the role of the mixed volume V₁(B, K) in [7] (up to dimensional factors). In addition to the volume functional considered in [7], we now treat general intrinsic volumes.

All these differences require a number of changes and new arguments, but also some parallel

reasoning is possible. In the latter cases, we will be brief and just list how the arguments of [7] have to be modified.

In analogy to Kendall’s original conjecture, of which a more general version was treated in [7], we have formulated the Corollary after Theorem 1. We make this more precise. As a space of ‘shapes’, suitable for our purpose, we may take the spaceS (with the Hausdorff metric) of convex bodies containingoand with circumradius one. The dilated version ofZ contained in Sis denoted byZ^∗. The conditional law for the shape ofZ, given the lower boundaforvk(Z), can be defined as the probability measure onS given byµ_a(·) := P(Z^∗ ∈ · |v_k(Z)≥a). We are asserting that lima→∞µ_a=δ_Bd weakly, whereδ_Bd is the Dirac measure onSconcentrated at the ballB^d(as we are prescribing the centre of the ball, this is a slightly stronger assertion than formulated in the Corollary). Thus, we have to prove that

lim sup

a→∞ µa(C)≤δ_B^d(C) (6)

for every closed setC ⊂ S. This is trivial ifB^d∈ C. LetB^d∈ C. Since the deviation measure/ ϑis continuous and positive onC, there exists >0 such that C ⊂ {K ∈ S :ϑ(K)≥}. This implies

µ_a(C)≤µ_a({K ∈ S :ϑ(K)≥}) = P(ϑ(Z)≥|v_k(Z)≥a)→0 fora→ ∞, by (1). This proves the Corollary.

Conversely, the assertion of the Corollary implies (1): given > 0, let C := {K ∈ S : ϑ(K)≥}, thenC is closed andB^d∈ C, hence (6) yields (1)./

As the details of the proofs for Theorems 1 – 3 are a bit technical, we want to sketch here the main lines of the reasoning, taking Theorem 1 as an example. Letk∈ {1, . . . , d}. In the first instance, we are interested in bounding the probability P(ϑ(Z) ≥ | v_k(Z) ≥ a) from above, for given >0 and largea. This conditional probability is a quotient. A lower bound for the denominator P(vk(Z)≥a) follows immediately from (5): the ballBa:= (a/κd)^1/kB^d satisfiesv_k(Ba) =a, hence

P(vk(Z)≥a)≥P(Y(H_Ba) = 0) = exp n

−2^dU(Ba)λ o

= exp n

−2^dκ^1−d/k_d a^d/kλ o

. (7) To obtain an upper bound for the numerator P(ϑ(Z)≥, v_k(Z)≥a) , we use the geometric inequality

U(K)≥κ^1−d/k_d v_k(K)^d/k, (8)

in a strengthened form. Equality in (8) holds if and only if K is a ball with centre o. Let K∈ K^d_o be a convex body satisfying ϑ(K)≥ >0. It can be proved (Lemma 1) that

U(K)≥(1 +f())κ^1−d/k_d v_k(K)^d/k withf()>0. If nowK satisfies

ϑ(K)≥ and v_k(K)≥a, (9)

this gives

P(Y(H_K) = 0) = exp n

−2^dU(K)λ o

≤exp n

−(1 +f())2^dκ^1−d/k_d a^d/kλ o

.

Since a convex body contained in the interior of the cellZ does not meet any hyperplane of Y, one might now hope that this estimate remains essentially true if the fixed convex bodyK

is replaced by the cellZ, in the cases where the latter satisfies the inequalities (9). Although this replacement is not legitimate, a similar and slightly weaker inequality of the form

P(ϑ(Z)≥, v_k(Z)≥a)≤c exp n

−(1 +g())2^dκ^1−d/k_d a^d/kλ o

, (10)

with a constantc > 0 and g() > 0, might be true. If this holds, then dividing (10) by (7) immediately gives

P(ϑ(Z)≥|v_k(Z)≥a)≤cexpn

−g()2^dκ^1−d/k_d a^d/kλo , which is of the required type.

An estimate of type (10) can indeed be proved, but at first not for v_k(Z) in intervals [a,∞), but in intervalsa(1,2). This is done in Lemmas 4 and 6. The distinction of two cases is necessary since it turns out that cells which are in a sense ‘too elongated’ need an extra treatment. The convex bodies are classified by an integer parameterm such that increasing m means increasing elongation. The auxiliary geometric Lemma 3 provides inner and outer inclusion estimates for bodies of given elongation. For largem, the estimate of Lemma 4 can be used; here the condition ϑ(Z) ≥ does not play a role. Its proof uses only elementary geometric arguments. For small m, the condition ϑ(Z) ≥ is essential. Lemma 6 contains the relevant estimate. It is here that the geometric stability result of Lemma 1 is needed.

Moreover, the approximation result for polytopes expressed in Lemma 5 is required for the reduction to a situation involving only a fixed number of hyperplanes. The further Lemmas 7, 8, 9 are needed to pass from intervals a(1,2) to intervals a(1,1 +h), with fixed h; the extension is achieved by a transformation. The obtained estimates are then combined into Lemma 10, which gives an upper estimate for the probability P(ϑ(Z)≥, v_k(Z)∈a(1,1+h)).

Since this upper bound contains h as a factor, it is necessary to estimate the denominator P(vk(Z) ∈ a(1,1 + h)) from below by a suitable bound which is also linear in h. This is achieved in Lemma 2. Its proof is essentially constructive, exhibiting sufficiently many realizations ofY for which the eventv_k(Z)∈a(1,1 +h) occurs. In both, Lemmas 2 and 10, the numberh must be sufficiently small. The final proof of Theorem 1 extends the estimates from the special intervalsa(1,1 +h), with small h, to general intervals [a, b), by a covering argument, as in the proof of Theorem 1 in [7].

3 A stability estimate

ForK ∈ K^d_o, we trivially have K ⊂Φ(K), henceU(K)≥vd(K). Here equality holds if and only ifK is a ball with centreo(this follows from the considerations below, but can also be shown directly). A similar inequality can be obtained for the other intrinsic volumes. In the following, we writeh(K,·) =:h_K for the support function ofK. Integrations with respect to σ extend overS^d−1. By H¨older’s inequality,

U(K)≥ 1

d(dκ_d)^1−d Z

h_Kdσ d

,

and since Z

hKdσ=dv1(K), we get

U(K)^1/d≥κ^(1−d)/d_d v1(K).

A well-known inequality (e.g., [15, p. 334]) says that

v₁(K)^k ≥κ^k−1_d v_k(K) (11)

fork= 1, . . . , d. Hence,

U(K)≥κ^1−d/k_d v_k(K)^d/k. (12)

Equality for a numberk∈ {1, . . . , d} holds if and only if K is a ball with centreo. We will need an improved version of (12), in the form of a stability estimate. The following proof combines techniques developed for obtaining stability results related to H¨older’s inequality and to isoperimetric inequalities such as (11).

Lemma 1. For K∈ K^d_o andk∈ {1, . . . , d},

U(K)≥(1 +γϑ(K)^(d+3)/2)κ^1−d/k_d v_k(K)^d/k, (13) where γ is a positive constant depending only on the dimension d.

Proof. We assume thatK contains more than one point; otherwise the assertion is trivial. In order to improve H¨older’s inequality, we use Lemma 4.2 of Gardner and Vassallo [2]. There we putm= 2, f0=f2 = 1,f1 =hK,w1= 1/d,w2 = (d−1)/d(hencew= 1/d), and obtain

1−

R hKdσ R h^d_Kdσ1/d R

1 dσ(d−1)/d

≥ 1 d

Z "

h^d/2_K

R h^d_Kdσ1/2 − 1 R 1 dσ1/2

#2

dσ =:β(K).

Using (1−β)(1 +β)≤1, we deduce that

U(K)^1/d≥(1 +β(K))κ^(1−d)/d_d v₁(K). (14) Next, we establish an estimate of the form β(K) ≥cϑ(K)^α with α >0. For this, we argue similarly as in the proof of Lemma 1 in [5] (which is reproduced in [4], see inequality (2.3.3)).

From now on in this paper,c₁, c₂, . . . denote constants depending only on the dimension, except in those cases where other dependences are explicitly indicated.

LetK ∈ K_o^dbe given. Sinceβ andϑare invariant under dilatations, we can normalizeK

and assume that Z

h^d_Kdσ = Z

1 dσ=dκd, (15)

which permits us to obtain the following relations. From (12) we get κ_d=U(K)≥κ^1−d_d v₁(K)^d≥c₁D(K)^d, whereDdenotes the diameter, hence D(K)≤c2 and therefore

h_K ≤c₂. (16)

Moreover,

β(K) =c₃ Z

h^d/2_K −12

dσ.

Suppose that ρoB^d ⊂ K ⊂RoB^d, where ρo is maximal and Ro is minimal. It follows from (15) thatρ_o≤1≤R_o.

Case 1: R_o−1≥1−ρ_o. We putRo = 1 +h, then

ϑ(K) = Ro−ρo

R_o+ρ_o ≤2h. (17)

There exists a vectoru0 ∈S^d−1 such thathK(u0) =Ro = 1 +h, and the pointp= (1 +h)u0

belongs toK. Foru∈S^d−1, letE(u) be the hyperplane throughpand orthogonal tou. Let ω denote the angle between u and u₀, and let ω₀ be the angle between u₀ and any v such thatE(v) is tangent to B^d. Then cosω0 = 1/(1 +h) and (withσd−1 := (d−1)κd−1)

Z

h^d/2_K −12

dσ ≥σd−1

Z ω0

0

n

[(1 +h) cosω]^d/2−1o2

(sinω)^d−2dω.

We set (1 +h)^d/2−1 =:α and substitute [(1 +h) cosω]^d/2−1 =αx, to obtain Z

h^d/2_K −12

dσ

≥ 2σd−1α³ d(1 +h)^d−2

Z 1 0

x²[(1 +h)²−(αx+ 1)^4/d]^(d−3)/2(αx+ 1)^(2−d)/ddx.

By (16), we can estimate

(1 +h)^2−d≥c4

and, for 0≤x≤1,

(αx+ 1)^(2−d)/d ≥(α+ 1)^(2−d)/d= (1 +h)^(2−d)/2≥c^1/2₄ . This gives

β(K)≥c5α³ Z 1

0

x²[(1 +h)²−(αx+ 1)^4/d]^(d−3)/2dx.

The function

f(x) := (1 +h)²−(αx+ 1)^4/d, 0≤x≤1,

satisfiesf(0) =h(h+ 2),f(1) = 0, andf⁰(1) =−(4/d)α(1 +h)^(4−d)/2. It is convex ford≥4 and concave ford= 2,3. Ford≥4 we deduce that

f(x)≥ 4

dα(1 +h)^(4−d)/2(1−x)≥c6α(1−x), and ford= 2,3 we get

f(x)≥h(h+ 2)(1−x)≥h(1−x).

Together with

α= (1 +h)^d/2−1≥ d 2h this yields

β(K)≥c7h^(d+3)/2. From (17) we now get

β(K)≥c8ϑ(K)^(d+3)/2.

Case 2: Ro−1<1−ρo.

We putρ_o = 1−h, then R_o <1 +h, hence

K ⊂(1 +h)B^d (18)

and

ϑ(K)≤2h. (19)

There is a vectoru₀ ∈S^d−1 such thath_K(u₀) =ρ_o= 1−h, and the hyperplane Gthrough ρou0 orthogonal tou0 supportsK. Letp∈(∂(1 +h)B^d)∩G. Letu1 ∈S^d−1 be the vector, positively spanned byu0 and p, that is orthogonal to a support plane ofB^d through p. Let ω₀ be the angle betweenu₀ andu₁. Foru∈S^d−1, letω_u be the angle betweenu₀ and u. If ωu≤ω0, then hK(u)≤1, hence 1−hK(u)^d/2 ≥1−hK(u)≥0 and thus

β(K)≥c3

Z

(1−hK(u))²1{ω_u ≤ω0}σ(du).

Now exactly the argument ofCase 2in [5, pp. 71–72] leads to β(K)≥c9h^(d+3)/2, which together with (19) gives

β(K)≥c₁₀ϑ(K)^(d+3)/2.

In each case, we conclude from (14) and (11) the inequality (13).

4 Probabilities involving small intervals

In this section we prove an inequality which replaces the easily obtained estimate (7) in the case wherevk(Z) is contained in a bounded interval.

ForA⊂R^d we define

Z(A) := \

x∈A

H⁻(x)

and setZ(x1, . . . ,xn) := Z({x₁, . . . ,xn}). For the random polytopeZ( ˜X), the typical cell ofX, we retain the notationZ.

Letk∈ {1, . . . , d}be fixed.

A main feature of the following estimate is the linear dependence of the lower bound on the length h of the interval (1,1 +h). A similar result, for the zero cell of a stationary Poisson hyperplane process and the function vd, is given in [7] as Lemma 3.2. It is an important advantage of the present argument that it merely uses the monotoneity, continuity and homogeneity properties of the functionv_k.

Lemma 2. For eachβ >0, there are constants h0>0, N ∈N and c11>0, depending only onβ and d, such that for a >0 and 0< h < h₀,

P(vk(Z)∈a(1,1 +h))≥c11h

a^d/kλ N

exp n

−(1 +β)^d2^dκ^1−d/k_d a^d/kλ o

.

Proof. Letβ >0 and a >0 be given. Forn≥2 we define Q_n:=

n

(x1, . . . ,xn−1,u)∈

(1 +β/2)B^d n−1

×S^d−1:

Z(x₁, . . . ,xn−1,u)⊂2⁻¹(1 +β/2)B^d, v_k(Z(x₁, . . . ,xn−1,u))≥2^−kκ_do . A continuity argument shows that we can chooseN =N(β, d)∈N(sufficiently large) so that

Z

S^d−1

Z

R^d

. . . Z

R^d

1{(x₁, . . . ,xN−1,u)∈ Q_N}dx1. . .dxN−1σ(du) =:c12(d, β)>0.

If (x1, . . . ,xN−1,u)∈ Q_N, then

v_k(Z(x₁, . . . ,x_N−1,u))≤(1 +β/2)^k2^−kκ_d. (20) We defineh₀ =h₀(β)>0 by

(1 +h0)^1/k(1 +β/2) = 1 +β (21) and suppose that 0< h < h0. As before, we put Ba:= (a/κd)^1/kB^d.

We start with the trivial estimate (again we use that a realization of ˜X denotes a simple counting measure and also its support)

P(vk(Z)∈a(1,1 +h))

≥P

X(2(1 +˜ β)Ba) =N, Z( ˜X∩2(1 +β)Ba)⊂(1 +β)Ba, vk(Z( ˜X∩2(1 +β)Ba))∈a(1,1 +h)

= P

X(2(1 +˜ β)B_a) =N P

Z( ˜X∩2(1 +β)B_a)⊂(1 +β)B_a, v_k(Z( ˜X∩2(1 +β)Ba))∈a(1,1 +h)

X(2(1 +˜ β)Ba) =N

.

Since ˜X is a stationary Poisson process with intensity λ, we obtain (using Satz 3.2.3(b) of [16])

P(vk(Z)∈a(1,1 +h))

≥ λ^N

N! exp{−v_d(2(1 +β)B_a)λ}

× Z

R^d

. . . Z

R^d

1{∀i: x_i ∈2(1 +β)B_a}1{Z(x₁, . . . ,x_N)⊂(1 +β)B_a}

×1{v_k(Z(x₁, . . . ,x_N))∈a(1,1 +h)}dx₁. . .dx_N. Assume thatx₁, . . . ,x_N ∈R^dsatisfy the conditions

(i) (x₁, . . . ,x_N)∈ kx_NkQ_N;

(ii)v_k(Z(x₁, . . . ,x_N))∈a(1,1 +h).

Then, using (ii), (i) and the definition ofQ_N, we get a(1 +h)

kx_Nk^k ≥ vk(Z(x1, . . . ,xN))

kx_Nk^k ≥2^−kκd,

hence

kx_Nk ≤2(a/κ_d)^1/k(1 +h)^1/k. (22) By (21) and (22),

kx_Nk ≤2(1 +β)(a/κ_d)^1/k.

Further, using (i), the definition ofQ_N, (22) and (21), we find that, fori= 1, . . . , N−1, kx_ik ≤ kx_Nk(1 +β/2)≤2(a/κ_d)^1/k(1 +h)^1/k(1 +β/2)≤2(1 +β)(a/κ_d)^1/k, thusxi ∈2(1 +β)Ba. Finally, (i), the definition of Q_N, (22) and (21) imply that

Z(x₁, . . . ,x_N) ⊂ 2⁻¹kx_Nk(1 +β/2)B^d⊂(a/κ_d)^1/k(1 +h)^1/k(1 +β/2)B^d

⊂ (1 +β)B_a.

Hence, introducing polar coordinates, we obtain P(v_k(Z)∈a(1,1 +h))

≥ λ^N

N!exp{−v_d(2(1 +β)B_a)λ}

Z

R^d

. . . Z

R^d

1{(x₁, . . . ,x_N)∈ kx_NkQ_N}

×1{v_k(Z(x1, . . . ,xN))∈a(1,1 +h)}dx₁. . .dxN

= λ^N

N!exp{−v_d(2(1 +β)B_a)λ}

× Z

S^d−1

Z ∞ 0

Z

R^d

. . . Z

R^d

1{(x₁, . . . ,xN−1, ru)∈rQ_N}

×1{v_k(Z(x1, . . . ,xN−1, ru))∈a(1,1 +h)}r^d−1dx1. . .dxN−1dr σ(du).

Substitutingx_i =ry_i fori= 1, . . . , N −1, we get P(v_k(Z)∈a(1,1 +h))

≥ λ^N

N!exp{−v_d(2(1 +β)Ba)λ}

× Z

S^d−1

Z ∞ 0

Z

R^d

. . . Z

R^d

1{(y₁, . . . ,y_N−1,u)∈ Q_N}

×1{r^kv_k(Z(y₁, . . . ,y_N−1,u))∈a(1,1 +h)}r^{N d−1}dy₁. . .dy_N−1dr σ(du)

= λ^N

N!exp{−v_d(2(1 +β)Ba)λ}

× Z

S^d−1

Z

R^d

. . . Z

R^d

1{(y₁, . . . ,y_N−1,u)∈ Q_N}

× a^{N d/k}

N d vk(Z(y₁, . . . ,y_N−1,u))^{−N d/k}

(1 +h)^{N d/k}−1

dy₁. . .dy_N−1σ(du)

≥ λ^N

N!exp{−v_d(2(1 +β)B_a)λ}a^{N d/k} N d

2^k(1 +β/2)^−kκ⁻¹_d N d/k N d

k h c₁₂(d, β)

≥h

a^d/kλ N

exp{−v_d(2(1 +β)Ba)λ}c13(d, β),

where also (20) was used. Since vd(2(1 +β)Ba) = (1 +β)^d2^dκ^1−d/k_d a^d/k, this proves the assertion.

5 Probabilities involving elongated cells

We will later need estimates showing that typical cells which, compared to their value ofv_k, are ‘too long’, occur only with small probability. This requires the following preparations, in the course of which convex bodies are classified according to their degree of elongation.

We denote byP_o^d⊂ K^d_o the subset of convex polytopes and byG(d, k) the Grassmannian of k-dimensional linear subspaces of R^d. For K ∈ K^d_o and L ∈ G(d, k), the set K|L is the image ofK under orthogonal projection toL. Fork∈ {1, . . . , d}, we define

η_k(K) := min{D(K)/∆(K, L) :L∈G(d, k)},

where D(K) denotes the diameter of K and ∆(K, L) is the width of K|L evaluated in L.

The simplest cases arek=d, where η_d(K) is the ratio of the diameter and the width of K, andk= 1, where η₁(K) = 1.

Leta >0 be given. Form∈N, we set K^d,k_a (m) :=

n

K ∈ K^d_o :v_k(K)∈a(1,2), η_k(K)∈[m^k,(m+ 1)^k) o

.

The information that K ∈ K^d,k_a (m) can be used to derive estimates for the size of K from above and below.

Lemma 3. Let m∈Nand k∈ {1, . . . , d}. Then (a)K∈ K_a^d,k(m) implies that K⊂c14m^ka^1/kB^d=:C;

(b)there exists a measurable map K^d,k_a (m)∩ P_o^d3P 7→v(P) such that v(P) is a vertex of P withkv(P)k ≥c₁₅ma^1/k.

Proof. We use repeatedly that v_k(K|L) is a constant multiple of the k-dimensional volume ofK|L.

(a) A special case of equation (5.3.23) in [15] and the monotoneity of mixed volumes imply that

v_k(K|L)≤c16v_k(K)

holds for allK ∈ K^d_o and L∈G(d, k). Let K ∈ K^d,k_a (m), and chooseL₀ ∈G(d, k) such that ηk(K) =D(K)/∆(K, L0). Then

2a≥vk(K)≥c⁻¹₁₆vk(K|L₀)≥c17∆(K, L0)^k−1D(K|L₀), where the estimate (16) from [7] was used. Thus

2a≥c₁₇∆(K, L₀)^k ≥c₁₇(m+ 1)^−k2D(K)^k and therefore

D(K)≤c₁₈m^ka^1/k. Sinceo∈K, this implies (a).

(b) For any L ∈ G(d, k), we enclose K|L in a rectangular parallelepiped in L with one edge length equal to ∆(K, L) and the other edge lengths at mostD(K|L). Then

v_k(K|L)≤c19∆(K, L)D(K|L)^k−1 ≤c19m^−kD(K)D(K)^k−1

and hence, by an integral-geometric projection formula ([15], (5.3.27)), a≤vk(K)≤c20m^−kD(K)^k.

Therefore, K has a point at distance at least 2⁻¹c^−1/k₂₀ ma^1/k from the origin. If K is a polytope, such a point can be chosen as a vertex. That a measurable selection is possible, follows as in [7, Lemma 4.3(c)].

Remark. We have η1(K) = 1; moreover, K^d,1_a (m) 6=∅ only for m= 1. Therefore, some of the subsequent arguments simplify considerably, or can be omitted, in the casek= 1.

Leta >0, >0 be given. Form∈N, we define

K_a,^d,k(m) :={K ∈ K_a^d,k(m) :ϑ(K)≥} (23) and

q^k_a,(m) := P(Z ∈ K^d,k_a,(m)). (24) Similarly as in [7], we prove two estimates concerning the decay of q^k_a,(m) as a^d/kλ→ ∞.

The dependence onwill not play a role until Lemma 6.

Foru1, . . . ,un∈S^d−1 and t1, . . . , tn∈(0,∞) we introduce the abbreviation

n

\

i=1

H⁻(u_i, t_i) =:P(u_(n), t_(n)).

Our next lemma corresponds to Lemma 5.1 in [7]; its proof is a modified and slightly simplified version of the proof of the latter.

Lemma 4. For m∈N and a^d/kλ≥σ0, where σ0 >0 is a constant,

q_a,^k (m)≤c₂₁(d, σ₀) exp{−c₂₂m^da^d/kλ}. (25) Proof. LetC be the ball defined in Lemma 3(a). Then

q^k_a,(m) =

∞

X

N=d+1

P(Y(H_C) =N)P(Z ∈ K^d,k_a,(m)|Y(H_C) =N). (26) Here,

pN := P(Z ∈ K^d,k_a,(m)|Y(H_C) =N)

= 1

U(C)^N Z

S^d−1

Z ∞ 0

. . . Z

S^d−1

Z ∞ 0

1 n

P(u_(N), t_(N))∈ K^d,k_a,(m) o

(27)

×1{∀i: H(u_i, t_i)∩C6=∅}(t₁· · ·t_N)^d−1dt₁σ(du₁). . .dt_Nσ(du_N).

Suppose that u₁, . . . ,u_N, t₁, . . . , t_N are such that the indicator functions occurring in the multiple integral are all equal to one; thenP :=P(u_(N), t_(N))∈ K^d,k_a,(m) has a vertexv(P)

according to Lemma 3(b). This vertex is the intersection ofdfacets ofP. Hence, there exists an index setJ ⊂ {1, . . . , N} withdelements such that

{v(P)}= \

i∈J

H(u_i, t_i).

The segmentS := [o,v(P)] satisfies

relintS∩H(u_j, t_j) =∅ forj∈ {1, . . . , N} \J, where relint denotes the relative interior. SinceS⊂C, we have

Z

S^d−1

Z ∞ 0

1{H(u, t)∩C6=∅, H(u, t)∩S =∅}t^d−1dt σ(du)

=U(C)−U(S) =U(C)−2^−dκ_d|S|^d, (28)

where|S|denotes the length ofS. Similarly as in the proof of [7, Lemma 5.1] we obtain pN ≤

N d

1 U(C)^N

Z

S^d−1

Z ∞ 0

. . . Z

S^d−1

Z ∞ 0

1{H(u_i, ti)∩C6=∅, i= 1, . . . , d}

×[U(C)−c₂₃m^da^d/k]^N−d(t₁· · ·t_d)^d−1dt₁σ(du₁). . .dt_dσ(du_d)

= N

d

1− c23m^da^d/k U(C)

!N−d

. (29)

This leads to the estimate q_a,^k (m) ≤

∞

X

N=d+1

[2^dU(C)λ]^N

N! exp{−2^dU(C)λ}

N d

1−c₂₃m^da^d/k U(C)

!N−d

= 1

d![2^dU(C)λ]^dexp{−2^dU(C)λ}

×

∞

X

N=d+1

1 (N−d)!

h

2^dU(C)λ−c24m^da^d/kλ iN−d

≤ 1

d![2^dU(C)λ]^dexp n

−c₂₄m^da^d/kλ o

≤ c25m^kd2

a^d/kλ d

exp n

−c₂₄m^da^d/kλ o

≤ c₂₇(d, σ₀) expn

−c₂₆m^da^d/kλo , which completes the proof.

The following result allows us to approximate a given convex polytopeP by a polytopeL⊂P with a restricted number of vertices such thatU(L) is not much smaller thanU(P).

For a polytope P, let extP be the set of vertices andf₀(P) the number of vertices of P.

Lemma 5. Let α >0 be given. There is a number ν ∈ N depending only on d and α such that the following is true. For P ∈ P_o^d there exists a polytope L = L(P) ∈ P_o^d satisfying

extL ⊂ extP, f0(L) ≤ ν, and U(L) ≥ (1−α)U(P). Moreover, there exists a measurable selectionP 7→L(P).

Proof. The following can be extracted from the proof of Lemma 4.2 in [7]. There exist numbers k₀ = k₀(d) and b₀ = b₀(d) such that the following is true. Let P ∈ P_o^d be a polytope and let P ⊂ RB^d, where R is minimal. Let k ≥ k₀. There is a measurable map P 7→L(P) such thatL =L(P) is the convex hull of at most (k+ 1)d vertices ofP, o∈L, andP ⊂L+κRB^d withκ=b₀k^−2/(d−1).

There is a unit vectorusuch thatR[o,u]⊂P and henceU(P)≥U(R[o,u]) = 2^−dκ_dR^d. By a suitable choice ofk, depending only ondandα, we can achieve thatκ≤1 and 4^dκ≤α.

Sinceh_L≤R, we get U(P)≤ 1

d Z

S^d−1

[h_L+κR]^ddσ≤U(L) +κ·2^dκ_dR^d≤U(L) + 4^dκU(P), thusU(L)≥(1−α)U(P), andν = (k+ 1)dis the required number.

For the probabilitiesq_a,^k , we now state another upper bound, which is based on the stability estimate in Lemma 1 and on the preceding approximation result.

Lemma 6. For m∈N, ∈(0,1) anda, λ >0, q^k_a,(m)≤c28(d, )m^kd2νexp

n

−

1 +c29^(d+3)/2

2^dκ^1−d/k_d a^d/kλ o

, where ν depends only on dand .

Proof. We define C as in Lemma 3(a) and use (26) and (27) again. Assume that u1, . . . ,uN, t1, . . . , tN are such that the indicator functions in (27) are all equal to one. Then, by Lemma 1,

U P(u_(N), t_(N))

≥(1 +γ^(d+3)/2)κ^1−d/k_d a^d/k. (30) Letα:=γ^(d+3)/2/(2 +γ^(d+3)/2); then (1−α)(1 +γ^(d+3)/2) = 1 +α. Putc30:=γ/(2 +γ);

thenα > c₃₀^(d+3)/2. By Lemma 5, there are ν =ν(d, ) vertices of P(u_(N), t_(N)) such that the convex hullL=L P(u_(N), t_(N)

of these vertices satisfies U(L)≥(1−α)U P(u_(N), t_(N))

. (31)

The inequalities (30) and (31) imply that

U(L)≥(1 +α)κ^1−d/k_d a^d/k. (32)

Now the same argument as in the proof of Lemma 5.2 in [7], with the obvious modifications, yields

P(Z ∈ K^d,k_a,(m)|Y(H_C) =N)[U(C)]^N

≤

dν

X

j=d+1

N j

j d

ν

h

U(C)−(1 +α)κ^1−d/k_d a^d/kiN−j

[U(C)]^j.

Here j denotes the number of hyperplanes generating the vertices ofL, and ^j_d

bounds the number of points of intersection of these hyperplanes; thus ^j_dν

estimates the possibilities to choose the vertices ofL. The probability that the otherN−j hyperplanes intersectingC do not meetLis given by [U(C)−U(L)]^N−jU(C)^−N+j, which is estimated using (32).

Inserting the inequality in (26), we can continue as in [7], finally using U(C) = (c14m^ka^1/k)^dκ_dand α > c30^(d+3)/2. This completes the proof.

6 Proof of Theorem 1

From now on, the proofs follow essentially the lines of those given in [7]. We will, therefore, state only the necessary lemmas in their modified forms and refer to the corresponding proofs in [7].

Leta >0 and∈(0,1) be given. Forh∈(0,1] andm∈Nwe define K_a,,h^d,k (m) :=n

K∈ K_o^d:v_k(K)∈a(1,1 +h), η_k(K)∈[m^k,(m+ 1)^k), ϑ(K)≥o

; (33) thus

P(vk(Z)∈a(1,1 +h), ϑ(Z)≥) =

∞

X

m=1

q^k,h_a,(m) with

q^k,h_a,(m) := P

Z ∈ K^d,k_a,,h(m)

. (34)

Moreover, we put

q^k,h_a,(m, n) := P

Z ∈ K^d,k_a,,h(m), fd−1(Z) =n

(35) forn∈N; here fd−1(P) denotes the number of facets of a polytope P. Then we have

P(vk(Z)∈a(1,1 +h), ϑ(Z)≥) =

∞

X

m=1

∞

X

n=d+1

q_a,^k,h(m, n).

Finally, we define

R^k,h_a,(m, n) :=

n

(u1, . . . ,un, t1, . . . , tn)∈(S^d−1)ⁿ×(0,∞)ⁿ: P(u_(n), t_(n))∈ K^d,k_a,,h(m), fd−1 P(u_(n), t_(n))

=n, H(u_i, t_i)∩C6=∅ fori= 1, . . . , no

, where the ballC is again defined as in Lemma 3(a), for the givena, , m.

Lemma 7. For m, n∈N, n≥d+ 1and h∈(0,1], q^k,h_a,(m, n) = (2^dλ)ⁿ

n!

Z

· · · Z

| {z }

R^k,ha,(m,n)

expn

−2^dU P(u_(n), t_(n)) λo

×(t₁· · ·tn)^d−1dt1. . .dtnσ(du1). . . σ(dun).

The proof is the same as that for Lemma 6.1 in [7], with the obvious necessary changes.

Lemma 8. Let w >0,h∈(0,1/2)and r ≥d−1. Then Z ^k

√1+h

1

x^rexp{−wx^d}dx

≤ c₂₀hw[1 + (exp{w/2} −1)⁻¹] Z ^k

√ 2 1

x^rexp{−wx^d}dx.

After the substitution x^d =y, one can imitate the proof of Lemma 6.2 in [7] to obtain the result.

The next (technical) lemma states that each bound forqa,^k,1 yields a bound forqa,^k,hwhich is linear inh. This should be compared to the bound in Lemma 2 which is also linear inh.

Lemma 9. For m∈N, h∈(0,1/2) and a^d/kλ≥σ0 >0,

q_a,^k,h(m)≤c31(d, σ0)h a^d/kλ m^dkq_a,^k,1(m).

Again, the proof is obtained by adapting the corresponding one from [7], namely that of Lemma 6.3. After applying Lemmas 7 and 8, we arrive at the inequality

q_a,^k,h(m, n) ≤ (2^dλ)ⁿ n!

Z

U(m,n)

t(ζ)^ndc₃₂hU(K(ζ, t(ζ)))λ

×

1 + expn

2^d−1U(K(ζ, t(ζ)))λo

−1−1

× Z ^k

√ 2 1

s^nd−1expn

−2^dU(K(ζ, t(ζ)))λs^do ds

×(t1· · ·tn−1)^d−1dt1. . .dtn−1σ(du1). . . σ(dun),

whereU(m, n),t(ζ) andK(·,·) are defined as in [7], with the obvious changes. Now we have to observe thatvk(K(ζ, t(ζ))) =a, henceK(ζ, t(ζ))∈ K^d,k_a,(m), which implies that

c33m^da^d/k ≤U(K(ζ, t(ζ)))≤c34m^kda^d/k, by Lemma 3. The estimation can now be completed as in [7].

The following lemma establishes an upper estimate for an unconditional probability.

Lemma 10. Let ∈(0,1), h∈(0,1/2) anda^d/kλ≥σ0 >0. Then P(v_k(Z)∈a(1,1 +h), ϑ(Z)≥)

≤c35(d, , σ0)hexpn

−

1 + (c29/2)^(d+3)/2

2^dκ^1−d/k_d a^d/kλo .

Here c29 is the constant appearing in Lemma 6. The proof of Lemma 10 follows the one of Proposition 7.1 in [7] and uses Lemmas 9, 4 and 6, in this order.

The choice (1 +β)^d= 1 + (c29/4)^(d+3)/2 in Lemma 2 immediately proves Theorem 1 with b=a(1 +h) in the case h ≤min(h0,1/2). As to arbitrary b≥a, we observe that Lemmas 2 and 10 have the same structure as Lemma 3.2 and Proposition 7.1, respectively, in [7];

they differ only by the values of some parameters. It is, therefore, clear that Theorem 1 now follows precisely in the same way as Theorem 1 of [7] was proved.

7 Proof of Theorem 2

In this section, Y denotes a stationary isotropic Poisson hyperplane process in R^d with intensityλ >0. For a convex bodyK⊂R^d and for n∈N0, we have

P(Y(H_K) =n) = [2κ⁻¹_d v1(K)λ]ⁿ

n! exp{−2κ⁻¹_d v1(K)λ}, (36) by [7, (4)], where B is now the ball with surface area 1, thus B = (dκ_d)^−1/(d−1)B^d. Let Zo

be the zero cell of the tessellation induced byY.

Lemma 11. Let k ∈ {1, . . . , d}. For each β > 0, there are constants h0 > 0, N ∈ N and c₃₆>0, depending only on β and d, such that for a >0 and 0< h < h₀,

P(v_k(Zo)∈a(1,1 +h))≥c36h

a^1/kλ N

exp n

−(1 +β)2κ^−1/k_d a^1/kλ o

.

Proof. In order to be able to essentially copy the proof of Lemma 2, we define a measureψ onR^d by

ψ(A) := 1 dκ_d

Z

S^d−1

Z ∞ 0

1_A(tu) dt σ(du)

for Borel setsA ⊂R^d. Let ˜X be the Poisson process in R^d with intensity measure λψ, and let Y⁰ be the hyperplane process defined by Y⁰ := {H(x) : x ∈ X}. Then˜ Y⁰ is a Poisson process in the spaceHof hyperplanes, and for a Borel setA ⊂ H we have

EY⁰(A) = λψ({x∈R^d:H(x)∈ A})

= 2λ

dκd

Z

S^d−1

Z ∞ 0

1A(H(u, t)) dt σ(du).

This shows that Y⁰ has the same intensity measure as Y. Since Y and Y⁰ are Poisson processes, they are stochastically equivalent. We can now repeat the proof of Lemma 2, where we replaceZ by Zo and dxbyψ(dx). Further, we observe thatψ(d(ru)) = (dκ_d)⁻¹dr σ(du) forkuk = 1 and ψ(d(ry)) = rψ(dy) for r >0. In the exponential, v_d(2(1 +β)B_a) (where B_a = (a/κ_d)^1/kB^d) has to be replaced by ψ(2(1 +β)B_a) = 2(1 +β)(a/κ_d)^1/k. With these changes, the proof of Lemma 2 yields the assertion of Lemma 11.

We will need a stability version of the inequality (11).

Lemma 12. There is a positive constant γ, depending only on the dimension d, such that for ∈(0,1)and every convex body K⊂R^d withr_d(K)≥, the inequality

v₁(K)≥

1 +γ^(d+3)/2

κ^1−1/k_d v_k(K)^1/k (37)

holds fork= 2, . . . , d.

Proof. Without loss of generality, we assume that K has interior points, mean width 2 (the same as the unit ball), and Steiner pointo. We putvi :=vi(K) fori= 0, . . . , d(v0=κd) and use the Aleksandrov-Fenchel inequalitiesv_i²≥vi−1v_i+1fori= 1, . . . , d−1 (see [15]). Theorem 6.6.6 and Lemma 6.6.5 of [15] provide an improvement of the first of these inequalities, namely

v₁²−v0v2 ≥c37δ(K, B^d)^(d+3)/2, whereδ is the Hausdorff distance. This can be rewritten as

v₁ v₀ ≥ v₂

v₁

1 + c₃₇

v₀v₂δ(K, B^d)^(d+3)/2

≥ v₂

v₁(1 +α)

withα:=c₃₈δ(K, B^d)^(d+3)/2, sincev₀v₂ ≤v₁² =c₃₉(v₁ being a constant multiple of the mean width). From

v₁

v₀ ≥(1 +α)v₂

v₁ ≥ · · · ≥(1 +α) v_k vk−1

we get

v₁ v₀

k−1

≥(1 +α)^k−1v_k v₁ and thus

v₁^k

v₀^k−1v_k ≥1 +c38δ(K, B^d)^(d+3)/2. Ifδ:=δ(K, B^d)≥1/2, then (since <1)

v₁^k

v₀^k−1v_k ≥1 +c₄₀^(d+3)/2.

Ifδ ≤1/2, letrB^d⊂K ⊂sB^dwherer is maximal andsis minimal. Then (by the definition of the Hausdorff metric)s≤1 +δ and r ≥1−δ ≥1/2, hence≤r_d(K) ≤(s/r)−1≤4δ and thus

v₁^k v₀^k−1vk

≥1 +c₄₁^(d+3)/2. Both cases together give

v₁ ≥

1 +c₄₂^(d+3)/21/k

v^1−1/k₀ v_k^1/k ≥

1 +c₄₃^(d+3)/2

κ^1−1/k₀ v^1/k_k .

Theorem 2 can now be proved in essentially the same way as Theorem 1, and we list only the necessary changes in Sections 5 and 6, in addition to those already mentioned in the proof of Lemma 11. Definitions (23) and (24) are replaced by

K_a,^d,k(m) :={K ∈ K_a^d,k(m) :r_d(K)≥}, q^k_a,(m) := P(Z_o∈ K^d,k_a,(m)).

Lemma 13. Let k ∈ {1, . . . , d}. For m ∈ N and a^1/kλ ≥ σ₀, where σ₀ > 0 is a given constant,

q_a,^k (m)≤c44(d, σ0) exp{−c₄₅ma^1/kλ}. (38)