3 A new formula of Blaschke-Petkantschin type

Gaussian polytopes: variances and limit theorems

Daniel Hug and Matthias Reitzner October 29, 2004

Abstract

The convex hull ofnindependent random points inR^dchosen according to the normal distribu- tion is called a Gaussian polytope. Estimates for the variance of the number ofi-faces and for the variance of thei-th intrinsic volume, of a Gaussian polytope inR^d,d∈N, are established by means of the Efron-Stein jackknife inequality and a new formula of Blaschke-Petkantschin type. These es- timates imply laws of large numbers for the number ofi-faces and for thei-th intrinsic volume of a Gaussian polytope asn→ ∞.

1 Introduction and statements of results

LetX₁, . . . , X_nbe a Gaussian sample inR^d(d∈N), i.e., independent random points chosen according to thed-dimensional standard normal distribution with mean zero and covariance matrix ¹₂I_d. Denote by Pn = [X1, . . . , Xn]the convex hull of these random points, and call Pn a Gaussian polytope. We are interested in geometric functionals such as volume, intrinsic volumes, and the number ofi-dimensional faces of Gaussian polytopes. Most of the previous investigations were concerned with expectations of such functionals. The starting point of this line of research is marked by a classical paper by R´enyi and Sulanke [21] in which the asymptotic behaviour of the expected number of vertices,Ef₀(P_n), ofP_nis determined in the plane, and thus also the expected number of edges,Ef₁(P_n), asntends to infinity. This result was generalized by Raynaud [20] who investigated the asymptotic behaviour of the mean number of facets,Efd−1(P_n), for arbitrary dimensions. Both results are only particular cases of the formula

Ef_i(P_n) = 2^d

√ d

d i+ 1

βi,d−1(πlnn)^d−12 (1 +o(1)), (1.1) wherei ∈ {0, . . . , d−1} andd ∈ N, asn → ∞. This follows, in arbitrary dimensions, from work of Affentranger and Schneider [2] and Baryshnikov and Vitale [3]. Heref_i(P_n)denotes the number of i-faces ofPn, and βi,d−1 is the internal angle of a regular (d−1)-simplex at one of itsi-dimensional faces. Recently, a more direct proof of (1.1) and some additional relations, which cannot be derived from [2] and [3], were given in [12]. However, it turned out to be difficult to extend these results to higher moments offi(Pn), and thus to prove limit theorems. An exception is the particular casei= 0, where Hueter [10], [11] states a Central Limit Theorem,

f0(Pn)−Ef0(Pn) pVarf₀(P_n)

−→ ND (0,1) (1.2)

AMS 2000 subject classifications. Primary 52A22, 60D05; secondary 60C05, 62H10.

Key words and phrases. Random points, convex hull,f-vector, intrinsic volumes, geometric probability, normal distribution, Gaussian sample, Stochastic Geometry, variance, law of large numbers, limit theorems.

asntends to infinity; here −→^D denotes convergence in distribution andN(0,1)is the (one-dimensional) normal distribution. The asymptotic behaviour of the variance is asserted to be of the form

Varf0(Pn) = ¯cd(lnn)^d−12 (1 +o(1)),

asn → ∞. Most probably it is difficult to establish such a precise limit relation for all f_i(P_n), i ∈ {1, . . . , d−1}. Our first result provides an upper bound for the order of the variance offi(Pn), for all i∈ {0, . . . , d−1}, which is of the same order.

Theorem 1.1. Letf_i(P_n)be the number ofi-dimensional faces of ad-dimensional Gaussian polytope Pn,d∈N. Then there exists a positive constantcd, depending only on the dimension, such that

Varf_i(P_n)≤c_d(lnn)^d−12 (1.3)

for alli∈ {0, . . . , d−1}.

Combining Chebyshev’s inequality and (1.3), we obtain P

|f_i(Pn)−Efi(Pn)|(lnn)^−d−12 ≥ε

≤ε⁻²(lnn)^−(d−1)Varfi(Pn)≤ε⁻²cd(lnn)^−d−12 , and thus the random variablef_i(P_n)satisfies a (weak) law of large numbers for alld∈N(the cased= 1 is trivial). In fact, the law forfi(Pn)(lnn)^−(d−1)/2converges in probability to the law concentrated at a constant.

Corollary 1.2. For d ∈ N andi ∈ {0, . . . , d−1}, the number of i-dimensional faces, f_i(P_n), of a Gaussian polytopePninR^dsatisfies

f_i(P_n)(lnn)^−d−12 −→ 2^d

√ d

d i+ 1

βi,d−1π^d−12 in probability asn→ ∞.

Mass´e [16] deduces a corresponding weak law of large numbers ford= 2andi= 0from Hueter’s central limit theorem (1.2).

Our method of proof also works for the volume and, more generally, the intrinsic volumes. Denote by V_i(P_n)thei-th intrinsic volume of the Gaussian polytopeP_n; hence, for instance,V_d(P_n)is the volume, 2V_d−1(P_n)is the surface area andV₁(P_n)is a multiple of the mean width ofP_n. The expected values of thei-th intrinsic volumes were investigated by Affentranger [1] who proved that

EVi(Pn) = d

i κ_d

κd−i

(lnn)ⁱ² (1 +o(1)), (1.4) fori ∈ {1, . . . , d}, asntends to infinity, whereκj denotes the volume of thej-dimensional unit ball.

The cased= 1which has been excluded in [1] can be checked directly. Relation (1.4) was expected to hold, since a result of Geffroy [6] implies that the Hausdorff distance betweenP_nand thed-dimensional ball of radius(lnn)^1/2and centred at the origin converges almost surely to zero. But it seems that (1.4) cannot be deduced directly from Geffroy’s result.

In the planar case, Hueter also states Central Limit Theorems forV₁(P_n)andV₂(P_n), V1(Pn)−EV1(Pn)

pVarV₁(P_n)

−→ ND (0,1) and V2(Pn)−EV2(Pn) pVarV₂(P_n)

−→ ND (0,1).

The variances suggested by Hueter are of the formVarVi(Pn) = ¹₂π^3/2(lnn)ⁱ (1 +o(1)).That her result cannot be correct can be seen from the following: if the stated asymptotic behaviour of the vari- ances were true, this would immediately implyP(V₁(P_n) ≤ 0)→ Φ(−√⁴

4π)andP(V₂(P_n) ≤ 0) → Φ(−√⁴

4π). But then the stated probabilities would be positive for largen, which obviously cannot hold.

In the next theorem we give an upper bound for the variances, for alli= 1, . . . , dandd∈N.

Theorem 1.3. LetV_i(P_n)be thei-th intrinsic volume of a Gaussian polytopeP_ninR^d,d ∈ N. Then there exists a positive constantc_d, depending only on the dimension, such that

VarV_i(P_n)≤c_d(lnn)ⁱ⁻³² (1.5)

for alli∈ {1, . . . , d}.

LetX₁, X₂, . . .be a sequence of independent random points which are identically distributed accord- ing to thed-dimensional normal distribution, and letP_n= [X₁, . . . , X_n]. Ford= 1, the quantityV₁(P_n) is the sample range ofX1, . . . , Xn. Although its distribution and moments can be expressed as multiple integrals (see [19, Chapter 8], [13, Chapter 14] and [15]), explicit values are not available in general. The asymptotic behaviour of VarV₁(P_n)ford= 1is deduced in Chapter 14 (see equation (14.100)) of [13], which yields (with the present normalization) thatVarV₁(P_n) = ^π₁₂²(lnn)⁻¹(1 +o(1))asn→ ∞. A different extension of the univariate sample range to higher dimensions is given by the largest interpoint distance of the given random points. Limiting distributions have been considered in [17] and in a more general framework in [8]. Still annother extension is discussed in [9].

From (1.4) and (1.5) we obtain an additive weak law of large numbers fori∈ {1,2}, i.e.

V_i(P_n)− d

i κ_d

κd−i

(lnn)²ⁱ →0

in probability as n → ∞. In order to derive a (multiplicative) strong law of large numbers for i ∈ {1, . . . , d}, we put n_k = 2^k. From the upper bound for the variance and Chebyshev’s inequality we deduce that

P

|V_i(Pn_k)−EVi(Pn_k)|(lnn_k)⁻ⁱ² ≥ε

≤ε⁻²c_d(lnn_k)⁻ⁱ⁺³² . Since

X

k≥1

(lnnk)^−(i+3)/2 = (ln 2)^−(i+3)/2X

k≥1

k^−(i+3)/2 <∞,

(1.4) and the Borell-Cantelli Lemma imply that

V_i(P_nk)(lnn_k)⁻ⁱ² → d

i κ_d

κd−i

(1.6) with probability one asktends to infinity. Moreover, sincen7→V_i(P_n)is increasing,

Vi(Pnk−1)(lnnk)⁻²ⁱ ≤Vi(Pn)(lnn)⁻ⁱ² ≤Vi(Pnk)(lnnk−1)⁻²ⁱ

fornk−1 ≤ n ≤ n_k, where by definition (lnn_k+1)/(lnn_k) → 1. Thus (1.6) implies a strong law of large numbers.

Corollary 1.4. LetVi(Pn)be thei-th intrinsic volume of a Gaussian polytopePninR^d,d∈N. Then, fori∈ {1, . . . , d},

V_i(P_n)(lnn)⁻²ⁱ → d

i κ_d

κd−i

with probability one asn→ ∞.

This law of large numbers can also be deduced from a result of Geffroy in [6].

The estimates for the variances obtained in Theorems 1.1 and 1.3 are based on the solution of another problem which is of independent interest. Consider the random polytopePn and choose another inde- pendent random pointX according to the normal distribution. The question we are interested in is the following: ifX /∈ Pn, how many facets ofPncan be seen fromX? We will determine the asymptotic behaviour of the expectation of the corresponding random variable, as n → ∞, and we will provide upper and lower bounds for its second moment.

In the following, let Fn(X)be the number of facets ofPnwhich can be seen fromX, i.e., which are up to(d−2)-dimensional faces contained in the interior of the convex hull ofPnandX. Note that F_n(X) = 0ifXis contained inP_n.

Theorem 1.5. LetX, X₁, . . . , X_n be independent random points inR^d, d ∈ N, which are identically distributed according to thed-dimensional normal distribution. LetP_d^(d−1)denote a Gaussian polytope inR^d−1. Then

n→∞lim EFn(X)n(lnn)^−d−12 = 2^d−1κdΓ(d+ 1)EVd−1(P_d^(d−1)). (1.7) Further, there is a positive constantcd, depending only on the dimension, such that

c⁻¹_d n⁻¹(lnn)^d−12 ≤EFn(X)² ≤cdn⁻¹(lnn)^d−12 . (1.8) For more information on random polytopes, we refer to the recent survey article by Schneider [24].

2 Projections of high-dimensional simplices

We want to give two interpretations of our results. The first one uses the fact that any orthogonal projec- tion of a Gaussian sample again is a Gaussian sample. So we make our notation more precise by writing P_n^(d)for a Gaussian polytope inR^dwhich is generated as the convex hull ofnnormally distributed ran- dom points inR^d. LetΠi :R^d→Rⁱbe the projection to the firsticomponents (i < d). For an arbitrary i-dimensional subspace ofR^d, which we identify withRⁱ, we then obtain

ϕ(Π_iP_n^(d))=^d ϕ(P_n⁽ⁱ⁾), (2.1)

where=^d means equality in distribution andϕis any (measurable) functional on the convex polytopes.

Now letP_n+1⁽ⁿ⁾ be a Gaussian simplex inRⁿ. As a consequence of Corollary 1.4 and (2.1), we obtain a law of large numbers for projections of high-dimensional random simplices: for a fixed integeri≥1,

Vi(ΠiP_n+1⁽ⁿ⁾) (lnn)⁻²ⁱ →κi

in probability asn→ ∞. Moreover, for a fixed integeri≥1, (1.4) implies that EVi(ΠiP_n+1⁽ⁿ⁾) =EVi(P_n+1⁽ⁱ⁾ ) =κi(lnn)²ⁱ (1 +o(1)),

asn→ ∞. An estimate of the variance can be deduced from Theorem 1.3. Thus, fori≥1, we have VarVi(ΠiP_n+1⁽ⁿ⁾)≤ci(lnn)ⁱ⁻³² .

Finally, Kubota’s theorem (see [25, (4.6)], [23, (5.3.27)]), H¨older’s inequality, and Theorem 1.3 yield the following asymptotic result for thei-th intrinsic volume of a high-dimensional Gaussian simplex (cf. the proof of Theorem 1.3 in Section 7 for a similar argument).

Corollary 2.1. LetV_i(P_n+1⁽ⁿ⁾) be thei-th intrinsic volume of a Gaussian simplex inRⁿ. Then, for any fixed integeri≥1,

V_i(P_n+1⁽ⁿ⁾)c⁻¹_n,i(lnn)⁻²ⁱ →κ_i in probability asn→ ∞, wherecn,i= ⁿ_i

κn/(κiκn−i).

Another method of generatingn+ 1random points inR^dgoes back to a suggestion of Goodman and Pollack. LetRdenote a random rotation ofRⁿ, putΠ^∗_d:= Π_d◦R, (recall thatΠ_ddenotes the projection onto R^d) and letT⁽ⁿ⁾ be a regular simplex in Rⁿ. ThenΠ^∗_d(T⁽ⁿ⁾) is a random polytope in R^d in the Goodman-Pollack model. It was proved by Baryshnikov and Vitale [3] that

ϕ(Π^∗_dT⁽ⁿ⁾)=^d ϕ(P_n+1^(d) ), (2.2)

for any affine invariant (measurable) functional ϕ on the convex polytopes. Thus, if f_i denotes the number ofi-faces, (1.1) is equivalent to

Ef_i(Π^∗_dT⁽ⁿ⁾) = 2^d

√ d

d i+ 1

βi,d−1(πlnn)^d−12 (1 +o(1))

asn → ∞, which is what was actually proved by Affentranger and Schneider [2]. By (2.2), the bound for the variance in Theorem 1.1 and the law of large numbers in Corollary 1.2 now give bounds and a law of large numbers, respectively, forΠ^∗_dT⁽ⁿ⁾, i.e.

Varfi(Π^∗_dT⁽ⁿ⁾)≤cd(lnn)^d−12 and, ford∈N,

fi(Π^∗_dT⁽ⁿ⁾)(lnn)^−d−12 −→ 2^d

√ d

d i+ 1

βi,d−1π^d−12 in probability, asntends to infinity.

For further information on the ‘Goodman-Pollack model’ and related work of Vershik and Sporyshev [27], we refer to [2].

3 A new formula of Blaschke-Petkantschin type

We work in the d-dimensional Euclidean spacesR^d with scalar producth·,·iand induced normk · k.

Thed-dimensional Lebesgue measure inR^dwill be denoted byλ_d. We writeS^d−1for the Euclidean unit sphere andσfor spherical Lebesgue measure (the dimension will be clear from the context). Recall that for pointsx1, . . . , xm ∈ R^d, the convex hull of these points is denoted by[x1, . . . , xm]. IfP ⊂R^dis a (convex) polytope, then we writeF_k(P)for the set of itsk-dimensional faces andf_k(P)for the number of thesek-faces, wherek ∈ {0, . . . , d}. Thek-dimensional Lebesgue measure in ak-dimensional flat E ⊂R^dis denoted byλE. Subspaces are endowed with the induced scalar product and norm. Finally, we writeΓ(·)for the Gamma function, especiallyΓ(n+ 1) =n!forn∈N.

An important tool in our investigations will be a new formula of Blaschke-Petkantschin type. The classical affine Blaschke-Petkantschin formula (see [22], II. 12. 3., [25],§6.1) states that

Z

R^d

· · · Z

R^d

f(x₁, . . . , x_d)

d

Y

j=1

dλ_d(x_j)

= Γ(d) Z

H^d_d−1

Z

H

· · · Z

H

f(x1, . . . , xd)λH([x1, . . . , xd])

d

Y

j=1

dλH(xj)d¯µ(H)

for any nonnegative measurable functionf : (R^d)^d → R. Here µ¯ is the motion invariant Haar mea- sure on the affine Grassmannian H^d_d−1 of hyperplanes inR^d normalized such that the measure of all hyperplanes hitting the Euclidean unit ball is equal todκd. Any hyperplaneH with0 ∈/ H can be pa- rameterized (uniquely) by one of its unit normal vectorsu ∈ S^d−1 and its distancet ≥ 0to the origin such thatH={y∈R^d:hy, ui=t}. Then we have

µ(·) =¯ Z

S^d−1

Z ∞ 0

1{tu+u^⊥∈ ·}dt dσ(u),

whereu^⊥denotes the(d−1)-dimensional subspace ofR^dtotally orthogonal tou.

The affine Blaschke-Petkantschin formula relates the d-dimensional volume elements dλ_d(x_j) of points x1, . . . , xd to the differential d¯µ(H) of a hyperplane H and the (d−1)-dimensional volume elementsdλH(xj)of pointsxj ∈ H,j = 1, . . . , d. Intuitively speaking, instead of choosingdrandom points inR^d, we first choose a random hyperplane and then, in a second step, we choosedrandom points in this hyperplane. More precisely, the corresponding transformation involves a Jacobian of the form [x1, . . . , x_d].

In this paper we need an analogous formula for two sets of points. The pointsx₁, . . . , x2d−kdeter- mine two hyperplanesH₁, H₂ which are the affine span ofx₁, . . . , x_dandx_d−k+1, . . . , x_2d−k, respec- tively. The following formula of Blaschke-Petkantschin type relates thed-dimensional volume elements dλ_d(x_j), j = 1, . . . ,2d−k, to the differentials d¯µ(H₁), d¯µ(H₂) of the hyperplanes H₁, H₂, to the (d−1)-dimensional volume elementsdλ_H1(x_j)anddλ_H2(x_l)of pointsx_j,j = 1, . . . , d−kandx_l, l =d+ 1, . . . ,2d−k, which are contained in exactly one hyperplane, and to the(d−2)-dimensional volume elementsdλ_H1∩H2(x_j)of pointsx_j,j =d−k+ 1, . . . , d, which are contained in both hyper- planes. Again such a transformation involves a Jacobian which takes into account the angle between the hyperplanes. This angle is defined as the angle between the normal vectors of the hyperplanes. Since we only consider the sinus of this angle, the choice of the orientation of the normal vectors need not be specified.

Lemma 3.1. Let0 ≤ k ≤ d−1, and letg : (R^d)^2d−k → Rbe a nonnegative measurable function.

Then

Z

R^d

· · · Z

R^d

g(x1, . . . , x2d−k)

2d−k

Y

j=1

dλ_d(xj) (3.1)

= Γ(d)² Z

H^d_d−1

Z

H^d_d−1

Z

H1

· · · Z

H1

Z

H1∩H2

· · · Z

H1∩H2

Z

H2

· · · Z

H2

g(x₁, . . . , x2d−k)

×λ_H1([x₁, . . . , x_d])λ_H2([xd−k+1, . . . , x2d−k]) (sinϕ)^−k

×

2d−k

Y

j=d+1

dλ_H2(x_j)

d

Y

j=d−k+1

dλ_H1∩H₂(x_j)

d−k

Y

j=1

dλ_H1(x_j)d¯µ(H₁)d¯µ(H₂),

wheresinϕdenotes the sinus of the angle betweenH₁andH₂.

Proof. The Blaschke-Petkantschin formula applied tox1, . . . , xdand Fubini’s theorem show that Z

R^d

· · · Z

R^d

g(x1, . . . , x2d−k)

2d−k

Y

j=1

dλ_d(xj)

= Γ(d) Z

H^d_d−1

Z

H1

· · · Z

H1

Z

R^d

· · · Z

R^d

g(x₁, . . . , x2d−k)λ_H1([x₁, . . . , x_d])

×

2d−k

Y

j=d+1

dλ_d(xj)

d

Y

j=1

dλ_H1(xj)d¯µ(H1).

We fixH₁ and set I(f) =

Z

H1

· · · Z

H1

Z

R^d

· · · Z

R^d

f(xd−k+1, . . . , x2d−k)

2d−k

Y

j=d+1

dλd(xj)

d

Y

j=d−k+1

dλH1(xj)

for nonnegative measurable functionsf : (R^d)^d→R.

An essential ingredient of our proof is a special case of a generalized linear Blaschke-Petkantschin formula due to Vedel Jensen and Kiˆeu [14] (see also [26, Theorem 5.6, p. 135]). For all nonnegative measurable functionshand for a (fixed) hyperplaneH, we thus have

Z

H

· · · Z

H

Z

R^d

· · · Z

R^d

h(y₁, . . . yd−1)

d−1

Y

j=k+1

dλ_d(y_j)

k

Y

j=1

dλ_H(y_j)

= Γ(d) Z

L^d_d−1

Z

H∩L

· · · Z

H∩L

Z

L

· · · Z

L

h(y₁, . . . yd−1)

×λ_L([0, y₁, . . . , yd−1]) (sinϕ)^−k

d−1

Y

j=k+1

dλ_L(y_j)

k

Y

j=1

dλH∩L(y_j)d¯ν(L),

whereϕdenotes the angle betweenHandL, and¯νis the rotation invariant Haar measure on the Grass- mannianL^d_d−1of(d−1)-dimensional linear subspaces inR^dwith total measuredκ_d/2.

Using a standard argument (cf., e.g., Schneider and Weil [25],§6.1) this immediately gives a gener- alized affine Blaschke-Petkantschin formula forI(f). PutH =H₁−x_2d−kandx_d−k+j−x_2d−k=y_j; then

I(f) = Z

R^d

Z

H

· · · Z

H

Z

R^d

· · · Z

R^d

f(y1+x2d−k, . . . , yd−1+x2d−k, x2d−k)

×

d−1

Y

j=k+1

dλ_d(y_j)

k

Y

j=1

dλ_H(y_j)dλ_d(x2d−k)

=Γ(d) Z

R^d

Z

L^d_d−1

Z

H∩L

· · · Z

H∩L

Z

L

· · · Z

L

f(y₁+x2d−k, . . . , yd−1+x2d−k, x2d−k)

×λL([0, y1, . . . , yd−1]) (sinϕ)^−k

×

d−1

Y

j=k+1

dλ_L(y_j)

k

Y

j=1

dλH∩L(y_j)d¯ν(L)dλ_d(x2d−k)

=Γ(d) Z

H^d_d−1

Z

H1∩H₂

· · · Z

H1∩H₂

Z

H2

· · · Z

H2

f(xd−k+1, . . . , x2d−k)

×λ_H2([xd−k+1, . . . , x2d−k]) (sinϕ)^−k

×

2d−k

Y

j=d+1

dλH2(xj)

d

Y

j=d−k+1

dλH1∩H2(xj)d¯µ(H2).

Settingf(xd−k+1. . . , x2d−k) = Γ(d)g(x1, . . . , x2d−k)λ_H1([x1, . . . , x_d])for fixedx1, . . . xd−k, one can easily complete the proof of the lemma.

4 Some auxiliary estimates

A random pointXinR^dis said to be normally distributed with positive definited×d-covariance matrix Σand mean0, i.e.,X∼N(0,Σ), if it is chosen according to the density

f_Σ(x) = 1

p(2π)^ddet Σ e^{−12xTΣ−1x}, x∈R^d.

For simplicity, we will exclusively consider the caseΣ = ¹₂Id. In this case we putfΣ(x) = φd(x), or simplyf_Σ(x) =φ(x)ifd= 1. The one-dimensional normal distributionN(0,¹₂)is given by

Φ(z) :=

z

Z

−∞

φ(x)dx= 1

√π

z

Z

−∞

e^−x2dx, z∈R.

The corresponding measures having these functions as densities with respect to the appropriate Lebesgue measure, will be denoted bydφ_d(x)instead ofφ_d(x)dx, etc.

We will repeatedly use the following well known asymptotic expansions concerning the density of the normal distribution:

Lemma 4.1. Letj≥0andγ >0. Then, ash1→ ∞,

∞

Z

h1

(h₂−h₁)^jφ(h₂)^γdh₂= Γ(j+ 1)

(2γ)^j+1 h^−(j+1)₁ φ(h₁)^γ 1 +O h⁻²₁ .

Lemma 4.2. Forn∈N,α, β ∈Randγ >0,

∞

Z

1

Φ(h1)^n−αh^β₁φ(h1)^γdh1= Γ(γ)2^γ−1n^−γ(lnn)

β+γ−1

2 (1 +o(1))

asn→ ∞.

The proof of Lemma 4.1 is immediate by the substitutiont= 2γh1(h2−h1). The proof of Lemma 4.2 is a direct generalization of an argument given by Affentranger [1].

We provide two useful estimates which will be needed later.

Lemma 4.3. Letj, l ≥0andγ >0. Then there exists a constantc > 0depending only onj, l, γ such that, forh₁ ≥1,

∞

Z

h1

π

Z

0

(h₂−h₁)^jφ(h₂)^γφ

h₁sinϕ 2

(sinϕ)^l dϕ dh₂ ≤c h^−(j+l+2)₁ φ(h₁)^γ.

Proof. Sincesinis symmetric with respect to ^π₂, by Fubini’s theorem and Lemma 4.1, we get

∞

Z

h1

π

Z

0

(h2−h1)^jφ(h2)^γφ

h1sinϕ 2

(sinϕ)^l dϕ dh2

≤2

∞

Z

h1

(h₂−h₁)^jφ(h₂)^γdh₂

π 2

Z

0

φ

h₁sinϕ 2

(sinϕ)^l dϕ

≤2c1h^−(j+1)₁ φ(h1)^γ

π

Z2

0

φ h1ϕ

π

ϕ^ldϕ,

where _π²ϕ≤sinϕ≤ϕfor0 ≤ϕ≤ ^π₂ was used in the last step. Here the constantc1depends only on j, γ. Substitutingh₁ϕ=t, we obtain

π

Z2

0

φ h₁ϕ

π

ϕ^ldϕ≤h^−(l+1)₁

∞

Z

0

φ t

π

t^ldt≤c₂h^−(l+1)₁ ,

wherec2is a constant depending only onl. Thus the assertion follows.

Lemma 4.4. Letj, l ≥0andγ >0. Then there exists a constantc > 0depending only onj, l, γ such that, forh₁ ≥1,

∞

Z

h1

π

Z

0

(h₂−h₁)^jφ(h₂)^γφ

h₁−h₂cosϕ 2 sinϕ

(sinϕ)^l−1 dϕ dh₂≤c h^−(j+l+1)₁ φ(h₁)^γ.

Proof. We will use Fubini’s theorem repeatedly and letc1, c2, . . .denote constants depending only on j, l, γ. Then, first substituting (for fixedh₂)

u= h1−h2cosϕ

sinϕ , du= h2−h1cosϕ sin²ϕ dϕ=

q

u²+h²₂−h²₁ sin⁻¹ϕ dϕ with

sinϕ= 1 u²+h²₂

h₁u+h₂ q

u²+h²₂−h²₁

, and thenh2 =h1+ _2h^s2

1, we get

∞

Z

h1

π

Z

0

(h2−h1)^jφ(h2)^γφ

h1−h2cosϕ 2 sinϕ

(sinϕ)^l−1 dϕ dh2

=

∞

Z

h1

∞

Z

−∞

(h2−h1)^jφ(h2)^γφ u

2

h1u+h2

pu²+h²₂−h²₁ l

pu²+h²₂−h²₁(u²+h²₂)^l du dh2

≤c1h^−j₁ φ(h1)^γ

∞

Z

−∞

∞

Z

0

φ(s)^γφ s²

2h₁ γ

φu 2

s^2jh^−2l₁

×

h₁u+

h₁+_2h^s2

1

qu²+s²+_4h^s42 1

l

q

u²+s²+_4h^s42 1

s h₁ds du

≤c2h^−(j+l+1)₁ φ(h1)^γ

∞

Z

−∞

∞

Z

0

φ(s)^γφ u

2

s^2j s

q

u²+s²+ _4h^s42 1

× |u|+

1 + s² 2h²₁

s

u²+s²+ s⁴ 4h²₁

!^l ds du

≤c2h^−(j+l+1)₁ φ(h1)^γ

∞

Z

−∞

∞

Z

0

φ(s)^γφ u

2 h

s^2j |u|+ (1 +s²)(|u|+s+s²)li ds du

≤c h^−(j+l+1)₁ φ(h₁)^γ, since the last double integral is finite.

5 Reduction of Theorem 1.1 to Theorem 1.5

The essential tool for estimating the variance of functionals of random polytopes is the Efron-Stein jackknife inequality [5], see also Efron [4] and Hall [7].

IfS =S(Y1, . . . , Yn)is any real symmetric function of the independent identically distributed ran- dom vectorsYj,1 ≤ j < ∞, we setSi = S(Y1, . . . , Yi−1, Yi+1, . . . , Yn+1)andS_(.) = _n+1¹ Pn+1

i=1 Si. The Efron-Stein jackknife inequality then says

VarS ≤E

n+1

X

i=1

(Si−S_(.))² = (n+ 1)E(Sn+1−S_(.))². (5.1) Note that the right-hand side is not decreased ifS_(.)is replaced by any other function ofY₁, . . . , Y_n+1.

We apply this inequality to the random variablef(Pn)wheref(·)is a measurable function of convex polytopes. ThenS =f([X₁, . . . , X_n]) = f(P_n), and we replaceS_(.)byf(P_n+1)which is a function of the convex hull ofP_nand a further random pointX_n+1. The Efron-Stein jackknife inequality then yields that

Varf(P_n)≤(n+ 1)E(f(P_n)−f(P_n+1))². (5.2) In the case thatf(·)is the number ofi-faces ofP_n, we obtain

Varfi(Pn)≤(n+ 1)E(fi(Pn+1)−fi(Pn))².

LetPnbe fixed and choose the additional random pointXn+1. If the pointXn+1 is contained inPn, the random variable fi(Pn+1)−fi(Pn) equals0. If Xn+1 ∈/ Pn, the relative interior of some of the i-dimensional faces ofP_n is contained in the interior of [P_n, X_n+1], let f_i⁻(X_n+1) be the number of theses faces, and some of thei-dimensional faces of[Pn, Xn+1]are not contained inPn, letf_i⁺(Xn+1) be the number of those faces. Then we have

|f_i([Pn, Xn+1])−fi(Pn)|=

f_i⁺(Xn+1)−f_i⁻(Xn+1)

≤f_i⁺(Xn+1) +f_i⁻(Xn+1).

SincePnis simplicial with probability one, this number can easily be estimated in terms of the number F_n(X_n+1)of facets ofP_nwhich can be seen fromX_n+1. HereF_n(X_n+1) = 0ifX_n+1is contained in P_n, and ifX_n+1 ∈/ P_nthen F_n(X_n+1) > 0is the number of facets ofP_nwhich are – up to(d−2)- dimensional faces – contained in the interior of the convex hull ofPnandXn+1. Now eachi-dimensional

“new” face of[P_n, X_n+1]not contained in P_n is the convex hull ofX_n+1 and an(i−1)-dimensional face ofP_n. Since this (i−1)-dimensional face is also a face of a facet ofP_n which can be seen from Xn+1, and each facet is a simplex, we obtain

f_i⁺(X_n+1)≤ d

i

F_n(X_n+1).

On the other hand eachi-dimensional face ofP_nwhich is – up to(i−1)-dimensional faces – contained in the interior of[P_n, X_n+1]is also a face of a facet contained in the interior of[P_n, X_n+1]. Hence

f_i⁻(X_n+1)≤ d

i+ 1

F_n(X_n+1) and combining these estimates proves

E(f_i(P_n+1)−f_i(P_n))²≤

d+ 1 i+ 1

2

EF_n(X_n+1)². (5.3)

Thus each estimate for the second moment ofF_n(X_n+1)yields an estimate forVarf_i(P_n), and hence

Theorem 1.1 follows from Theorem 1.5.

6 Proof of Theorem 1.5

6.1 Asymptotic expansion of the expectation

We start with the proof of the first part of Theorem 1.5. The case d = 1is included by proper inter- pretations of the subsequent arguments. Choosen+ 1independent normally distributed random points X1, . . . Xn, XinR^d. The convex hull of the firstnpoints is a Gaussian polytopePn, and with probabil- ity onePnis simplicial. ForI ⊂ {1, . . . , n}with|I|=d, denote byF_I the convex hull of{X_i :i∈I}

which is a(d−1)-dimensional simplex. The affine hull ofF_I is denoted byH(F_I). With probability one, this affine hull is a hyperplane which dissectsR^dinto two (closed) halfspaces. The halfspace which contains the origin will be denoted byH0(F_I), the other byH+(F_I). The origin is contained in exactly one halfspace with probability one. In the following, we want to assume thatP_ncontains the origin. This happens with high probability, since by Wendel’s theorem [28]

P(0∈/ P_n) =O(n^d2⁻ⁿ),

an thus the condition thatP_ncontains the origin can be inserted by adding a suitable error term. More- over, we can also assume that the pointsX1, . . . , Xn, X are in general relative position (i.e. any subset of at mostd+ 1of these random points is affinely independent).

We are interested in the number of facets ofPnwhich can be seen from the additional random point X /∈Pn. Denote the set of these facets byF_n(X), i.e.,

F_n(X) =F(X₁, . . . , Xn;X)

={F_I : Pn⊂H0(FI), X ∈H+(FI), I ⊂ {1, . . . , n},|I|=d} .

Here we can defineF_n(X) as the empty set, if the origin is not contained in the interior ofPn or if the random points are not in general relative position. Similar definitions can be given for deterministic

points x1, . . . , xn andx in general relative position such that the convex hull ofx1, . . . , xn contains the origin. When applying Wendel’s theorem in considering EF_n(X), the error term is of the order n^2d2⁻ⁿ=O(c⁻ⁿ)with a suitable constantc >1, sinceF_n(X)is bounded by ⁿ_d

. Using this we have EF_n(X) =

Z

R^d

· · · Z

R^d

X1{F_I ∈ F_n(x)}

n

Y

j=1

dφ_d(x_j)dφ_d(x) +O(c⁻ⁿ), (6.1)

where the summation extends over all subsetsI ⊂ {1, . . . , n}with|I| = d. Denote byF1 the convex hull ofx1, . . . , x_d. Then

EF_n(X) = n

d Z

R^d

· · · Z

R^d

1{F₁ ∈ F_n(x)}

n

Y

j=1

dφ_d(x_j)dφ_d(x) +O(c⁻ⁿ).

The probability content of the halfspaceH+(F1)is Z

H+(F1)

dφ_d(x) = 1−Φ(h1),

whereh₁ is the distance ofH(F₁) to the origin. IfF₁ ∈ F_n(X), thenX is contained in the halfspace H+(F1)with probability content1−Φ(h₁), and the random pointsXj,j∈ {d+1, . . . , n}, are contained in the halfspaceH₀(F₁)with probability contentΦ(h₁). Hence we obtain

EFn(X) = n

d Z

R^d

· · · Z

R^d

Φ(h1)^n−d(1−Φ(h1))

d

Y

j=1

dφd(xj) +O(c⁻ⁿ).

Parameterizing the hyperplaneH(F1) =:H1by its distanceh1 ≥0from the origin and its unit normal vectoru₁in the formH₁=H(u₁, h₁), and using the affine Blaschke-Petkantschin formula, we find that

EF_n(X) =Γ(d) n

d Z

S^d−1

∞

Z

0

Φ(h₁)^n−d(1−Φ(h₁))

×





 Z

H1

· · · Z

H1

λ_H1([x₁, . . . , x_d])

d

Y

j=1

(φ_d(x_j)dλ_H1(x_j))







dh₁dσ(u₁) +O(c⁻ⁿ).

The inner integral (in brackets) is the expected volumeEVd−1(P_d^(d−1))of a random(d−1)-dimensional Gaussian simplex inR^d−1timesφ(h₁)^d, which gives

EFn(X) = Γ(d) n

d

EVd−1(P_d^(d−1)) Z

S^d−1

∞

Z

0

Φ(h1)^n−d(1−Φ(h1))φ(h1)^ddh1dσ(u1) +O(c⁻ⁿ).

The expected volume of a random Gaussian simplex was computed explicitly by Miles [18]. It now follows from Lemma 4.1 and Lemma 4.2 that

EFn(X) = 2^d−1κ_dΓ(d+ 1)EVd−1(P_d^(d−1))n⁻¹(lnn)^d−12 (1 +o(1)). (6.2) This proves the first part of Theorem 1.5.

6.2 Estimate of the variance

The main part of the proof is devoted to estimating the second moment ofF_n(X). In the cased= 1, we haveF_n(X) =F_n(X)², hence the assertion follows. Now letd≥2.

As in (6.1) we have EF_n(X)² =

Z

R^d

· · · Z

R^d

X

I

1{F_I ∈ F_n(x)}

!2 n

Y

j=1

dφ_d(x_j)dφ_d(x) +O(c⁻ⁿ)

with somec >1. The summation extends over all subsetsI ⊂ {1, . . . , n}with|I|=d. We expand the integrand and get

EFn(X)²=X

I

X

J

Z

R^d

· · · Z

R^d

1{F_I, F_J ∈ F_n(x)}

n

Y

j=1

dφ_d(xj)dφ_d(x) +O(c⁻ⁿ), (6.3) where the summation extends over all subsets I, J ⊂ {1, . . . , n}with |I| = |J| = d. If we fix the numberk =|I ∩J| ∈ {0, . . . , d}, then the corresponding term in (6.3) depends only onkand not on the particular choice ofI and J. For given k ∈ {0, . . . , d}, we putF1 = [X1, . . . , X_d]andF₂^(k) = [Xd−k+1, . . . , X2d−k]. Note that fork=dwe haveF1=F₂^(d). HenceEFn(X)²can be rewritten as

EFn(X)² =

d

X

k=0

n d

d k

n−d d−k

Z

R^d

· · · Z

R^d

1{F₁, F₂^(k)∈ F_n(x)}

n

Y

j=1

dφ_d(xj)dφ_d(x) +O(c⁻ⁿ).

The summand corresponding tok=dis justEF_n(X), and thus (6.2) yields EFn(X)²≤c1

d−1

X

k=0

n^2d−k Z

R^d

· · · Z

R^d

1{F₁, F₂^(k)∈ F_n(x)}

n

Y

j=1

dφ_d(xj)dφ_d(x)

+O

n⁻¹(lnn)^d−12

;

here and in the followingc1, c2, . . .denote constants which are independent ofn. The summand corre- sponding tok=dalso yields the asserted lower bound forEFn(X)².

Leth₁, h₂be the distance to the origin andu₁, u₂the unit normal vector ofH(F₁), H(F₂^(k)), respec- tively, such thatH(F1) =H(u1, h1)andH(F₂^(k)) =H(u2, h2). Since the integrand is symmetric inF1

andF₂^(k), we restrict our integration toh1 ≤h2. Thus we get EFn(X)² ≤c2

d−1

X

k=0

n^2d−k Z

R^d

· · · Z

R^d

1{h₁ ≤h2}1{F₁, F₂^(k)∈ F_n(x)}

n

Y

j=1

dφd(xj)dφd(x)

+O

n⁻¹(lnn)^d−12

.

IfF1, F₂^(k) ∈ F_n(x), then the points x2d−k+1, . . . , xn are contained in H0(F1)∩H0(F₂^(k)), and the corresponding measure of the set of these points is at mostΦ(h1)^n−2d+k. Moreover,F1, F₂^(k)∈ F_n(x) implies thatxis contained inH₊(F₁)∩H₊(F₂^(k)). Denote the distance ofH₊(F₁)∩H₊(F₂^(k))to the origin byh12. Then the corresponding measure is at most1−Φ(h12). This yields

EF_n(X)²≤c₂

d−1

X

k=0

n^2d−k Z

R^d

· · · Z

R^d

1{h₁ ≤h₂}Φ(h₁)^n−2d+k(1−Φ(h₁₂))

2d−k

Y

j=1

dφ_d(x_j)

+O

n⁻¹(lnn)^d−12 .