Gaussian polytopes: variances and limit theorems
Daniel Hug and Matthias Reitzner
October 29, 2004
Abstract
The convex hull of n independent random points in Rd chosen according to the normal distribution is called a Gaussian polytope. Estimates for the variance of the number of i-faces and for the
variance of the i-th intrinsic volume, of a Gaussian polytope in Rd , d ∈ N, are established by means
of the Efron-Stein jackknife inequality and a new formula of Blaschke-Petkantschin type. These estimates imply laws of large numbers for the number of i-faces and for the i-th intrinsic volume of a
Gaussian polytope as n → ∞.
1
Introduction and statements of results
Let X1 , . . . , Xn be a Gaussian sample in Rd (d ∈ N), i.e., independent random points chosen according
to the d-dimensional standard normal distribution with mean zero and covariance matrix 21 Id . 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 of i-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 Rényi and
Sulanke [21] in which the asymptotic behaviour of the expected number of vertices, Ef0 (Pn ), of Pn is
determined in the plane, and thus also the expected number of edges, Ef1 (Pn ), as n tends to infinity. This
result was generalized by Raynaud [20] who investigated the asymptotic behaviour of the mean number
of facets, Efd−1 (Pn ), for arbitrary dimensions. Both results are only particular cases of the formula
d−1
2d
d
Efi (Pn ) = √
βi,d−1 (π ln n) 2 (1 + o(1)),
(1.1)
d i+1
where i ∈ {0, . . . , d − 1} and d ∈ N, as n → ∞. This follows, in arbitrary dimensions, from work
of Affentranger and Schneider [2] and Baryshnikov and Vitale [3]. Here fi (Pn ) denotes the number of
i-faces of Pn , and βi,d−1 is the internal angle of a regular (d − 1)-simplex at one of its i-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 of fi (Pn ), and thus to prove limit theorems. An exception is the particular case i = 0, where
Hueter [10], [11] states a Central Limit Theorem,
f0 (Pn ) − Ef0 (Pn ) D
p
−→ N (0, 1)
Var f0 (Pn )
(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.
1
D
as n tends to infinity; here −→ denotes convergence in distribution and N (0, 1) is the (one-dimensional)
normal distribution. The asymptotic behaviour of the variance is asserted to be of the form
Var f0 (Pn ) = c̄d (ln n)
d−1
2
(1 + o(1)),
as n → ∞. Most probably it is difficult to establish such a precise limit relation for all fi (Pn ), i ∈
{1, . . . , d − 1}. Our first result provides an upper bound for the order of the variance of fi (Pn ), for all
i ∈ {0, . . . , d − 1}, which is of the same order.
Theorem 1.1. Let fi (Pn ) be the number of i-dimensional faces of a d-dimensional Gaussian polytope
Pn , d ∈ N. Then there exists a positive constant cd , depending only on the dimension, such that
Var fi (Pn ) ≤ cd (ln n)
d−1
2
(1.3)
for all i ∈ {0, . . . , d − 1}.
Combining Chebyshev’s inequality and (1.3), we obtain
d−1
d−1
P |fi (Pn ) − Efi (Pn )|(ln n)− 2 ≥ ε ≤ ε−2 (ln n)−(d−1) Var fi (Pn ) ≤ ε−2 cd (ln n)− 2 ,
and thus the random variable fi (Pn ) satisfies a (weak) law of large numbers for all d ∈ N (the case d = 1
is trivial). In fact, the law for fi (Pn )(ln n)−(d−1)/2 converges in probability to the law concentrated at a
constant.
Corollary 1.2. For d ∈ N and i ∈ {0, . . . , d − 1}, the number of i-dimensional faces, fi (Pn ), of a
Gaussian polytope Pn in Rd satisfies
d−1
2d
d
− d−1
2
fi (Pn )(ln n)
−→ √
βi,d−1 π 2
i
+
1
d
in probability as n → ∞.
Massé [16] deduces a corresponding weak law of large numbers for d = 2 and i = 0 from Hueter’s
central limit theorem (1.2).
Our method of proof also works for the volume and, more generally, the intrinsic volumes. Denote by
Vi (Pn ) the i-th intrinsic volume of the Gaussian polytope Pn ; hence, for instance, Vd (Pn ) is the volume,
2Vd−1 (Pn ) is the surface area and V1 (Pn ) is a multiple of the mean width of Pn . The expected values of
the i-th intrinsic volumes were investigated by Affentranger [1] who proved that
i
d κd
(1.4)
EVi (Pn ) =
(ln n) 2 (1 + o(1)),
i κd−i
for i ∈ {1, . . . , d}, as n tends to infinity, where κj denotes the volume of the j-dimensional unit ball.
The case d = 1 which 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 between Pn and the d-dimensional
ball of radius (ln n)1/2 and 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 for V1 (Pn ) and V2 (Pn ),
V1 (Pn ) − EV1 (Pn ) D
p
−→ N (0, 1)
Var V1 (Pn )
and
2
V2 (Pn ) − EV2 (Pn ) D
p
−→ N (0, 1).
Var V2 (Pn )
The variances suggested by Hueter are of the form Var Vi (Pn ) = 12 π 3/2 (ln n)i (1 + o(1)) . That her
result cannot be correct can be seen from the following: if the stated asymptotic
behaviour of the vari√
4
ances√were true, this would immediately imply P(V1 (Pn ) ≤ 0) → Φ(− 4π) and P(V2 (Pn ) ≤ 0) →
Φ(− 4 4π). But then the stated probabilities would be positive for large n, which obviously cannot hold.
In the next theorem we give an upper bound for the variances, for all i = 1, . . . , d and d ∈ N.
Theorem 1.3. Let Vi (Pn ) be the i-th intrinsic volume of a Gaussian polytope Pn in Rd , d ∈ N. Then
there exists a positive constant cd , depending only on the dimension, such that
Var Vi (Pn ) ≤ cd (ln n)
i−3
2
(1.5)
for all i ∈ {1, . . . , d}.
Let X1 , X2 , . . . be a sequence of independent random points which are identically distributed according to the d-dimensional normal distribution, and let Pn = [X1 , . . . , Xn ]. For d = 1, the quantity V1 (Pn )
is the sample range of X1 , . . . , 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 Var V1 (Pn ) for d = 1 is deduced in Chapter 14 (see equation (14.100)) of [13],
2
which yields (with the present normalization) that Var V1 (Pn ) = π12 (ln n)−1 (1 + o(1)) as n → ∞. 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 for i ∈ {1, 2}, i.e.
i
d κd
Vi (Pn ) −
(ln n) 2 → 0
i κd−i
in probability as n → ∞. In order to derive a (multiplicative) strong law of large numbers for i ∈
{1, . . . , d}, we put nk = 2k . From the upper bound for the variance and Chebyshev’s inequality we
deduce that
i+3
i
P |Vi (Pnk ) − EVi (Pnk )|(ln nk )− 2 ≥ ε ≤ ε−2 cd (ln nk )− 2 .
Since
X
(ln nk )−(i+3)/2 = (ln 2)−(i+3)/2
k≥1
X
k −(i+3)/2 < ∞,
k≥1
(1.4) and the Borell-Cantelli Lemma imply that
− 2i
Vi (Pnk )(ln nk )
d κd
→
i κd−i
(1.6)
with probability one as k tends to infinity. Moreover, since n 7→ Vi (Pn ) is increasing,
i
i
i
Vi (Pnk−1 )(ln nk )− 2 ≤ Vi (Pn )(ln n)− 2 ≤ Vi (Pnk )(ln nk−1 )− 2
for nk−1 ≤ n ≤ nk , where by definition (ln nk+1 )/(ln nk ) → 1. Thus (1.6) implies a strong law of
large numbers.
Corollary 1.4. Let Vi (Pn ) be the i-th intrinsic volume of a Gaussian polytope Pn in Rd , d ∈ N. Then,
for i ∈ {1, . . . , d},
d κd
− 2i
Vi (Pn )(ln n) →
i κd−i
with probability one as n → ∞.
3
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 polytope Pn and choose another independent random point X according to the normal distribution. The question we are interested in is the
following: if X ∈
/ Pn , how many facets of Pn can be seen from X? 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 of Pn which can be seen from X, i.e., which
are up to (d − 2)-dimensional faces contained in the interior of the convex hull of Pn and X. Note that
Fn (X) = 0 if X is contained in Pn .
Theorem 1.5. Let X, X1 , . . . , Xn be independent random points in Rd , d ∈ N, which are identically
(d−1)
distributed according to the d-dimensional normal distribution. Let Pd
denote a Gaussian polytope
d−1
in R . Then
d−1
(d−1)
lim EFn (X) n(ln n)− 2 = 2d−1 κd Γ(d + 1)EVd−1 (Pd
).
(1.7)
n→∞
Further, there is a positive constant cd , depending only on the dimension, such that
−1
c−1
d n (ln n)
d−1
2
≤ EFn (X)2 ≤ cd n−1 (ln n)
d−1
2
.
(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 projection of a Gaussian sample again is a Gaussian sample. So we make our notation more precise by writing
(d)
Pn for a Gaussian polytope in Rd which is generated as the convex hull of n normally distributed random points in Rd . Let Πi : Rd → Ri be the projection to the first i components (i < d). For an arbitrary
i-dimensional subspace of Rd , which we identify with Ri , we then obtain
d
ϕ(Πi Pn(d) ) = ϕ(Pn(i) ),
(2.1)
d
where = means equality in distribution and ϕ is any (measurable) functional on the convex polytopes.
(n)
Now let Pn+1 be a Gaussian simplex in Rn . 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 integer i ≥ 1,
i
(n)
Vi (Πi Pn+1 ) (ln n)− 2 → κi
in probability as n → ∞. Moreover, for a fixed integer i ≥ 1, (1.4) implies that
(n)
i
(i)
EVi (Πi Pn+1 ) = EVi (Pn+1 ) = κi (ln n) 2 (1 + o(1)),
as n → ∞. An estimate of the variance can be deduced from Theorem 1.3. Thus, for i ≥ 1, we have
(n)
Var Vi (Πi Pn+1 ) ≤ ci (ln n)
i−3
2
.
Finally, Kubota’s theorem (see [25, (4.6)], [23, (5.3.27)]), Hölder’s inequality, and Theorem 1.3 yield the
following asymptotic result for the i-th intrinsic volume of a high-dimensional Gaussian simplex (cf. the
proof of Theorem 1.3 in Section 7 for a similar argument).
4
(n)
Corollary 2.1. Let Vi (Pn+1 ) be the i-th intrinsic volume of a Gaussian simplex in Rn . Then, for any
fixed integer i ≥ 1,
(n)
− 2i
Vi (Pn+1 ) c−1
→ κi
n,i (ln n)
n
in probability as n → ∞, where cn,i = i κn /(κi κn−i ).
Another method of generating n + 1 random points in Rd goes back to a suggestion of Goodman and
Pollack. Let R denote a random rotation of Rn , put Π∗d := Πd ◦ R, (recall that Πd denotes the projection
onto Rd ) and let T (n) be a regular simplex in Rn . Then Π∗d (T (n) ) is a random polytope in Rd in the
Goodman-Pollack model. It was proved by Baryshnikov and Vitale [3] that
d
(d)
ϕ(Π∗d T (n) ) = ϕ(Pn+1 ),
(2.2)
for any affine invariant (measurable) functional ϕ on the convex polytopes. Thus, if fi denotes the
number of i-faces, (1.1) is equivalent to
d−1
d
2d
∗ (n)
βi,d−1 (π ln n) 2 (1 + o(1))
Efi (Πd T ) = √
d i+1
as n → ∞, 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, for Π∗d T (n) , i.e.
Var fi (Π∗d T (n) ) ≤ cd (ln n)
and, for d ∈ N,
d−1
fi (Π∗d T (n) )(ln n)− 2
d−1
2
d−1
2d
d
√
−→
βi,d−1 π 2
i
+
1
d
in probability, as n tends 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 spaces Rd with scalar product h· , ·i and induced norm k · k.
The d-dimensional Lebesgue measure in Rd will be denoted by λd . We write Sd−1 for the Euclidean unit
sphere and σ for spherical Lebesgue measure (the dimension will be clear from the context). Recall that
for points x1 , . . . , xm ∈ Rd , the convex hull of these points is denoted by [x1 , . . . , xm ]. If P ⊂ Rd is a
(convex) polytope, then we write Fk (P ) for the set of its k-dimensional faces and fk (P ) for the number
of these k-faces, where k ∈ {0, . . . , d}. The k-dimensional Lebesgue measure in a k-dimensional flat
E ⊂ Rd is denoted by λE . Subspaces are endowed with the induced scalar product and norm. Finally,
we write Γ(·) for the Gamma function, especially Γ(n + 1) = n! for n ∈ 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
Z
d
Y
· · · f (x1 , . . . , xd )
dλd (xj )
Rd
j=1
Rd
Z
Z
= Γ(d)
d
H
Hd−1
Z
···
f (x1 , . . . , xd ) λH ([x1 , . . . , xd ])
d
Y
j=1
H
5
dλH (xj ) dµ̄(H)
for any nonnegative measurable function f : (Rd )d → R. Here µ̄ is the motion invariant Haar mead
sure on the affine Grassmannian Hd−1
of hyperplanes in Rd normalized such that the measure of all
hyperplanes hitting the Euclidean unit ball is equal to dκd . Any hyperplane H with 0 ∈
/ H can be parameterized (uniquely) by one of its unit normal vectors u ∈ Sd−1 and its distance t ≥ 0 to the origin
such that H = {y ∈ Rd : hy, ui = t}. Then we have
Z
Z ∞
µ̄(·) =
1{tu + u⊥ ∈ ·}dt dσ(u),
Sd−1
0
where u⊥ denotes the (d − 1)-dimensional subspace of Rd totally orthogonal to u.
The affine Blaschke-Petkantschin formula relates the d-dimensional volume elements dλd (xj ) of
points x1 , . . . , xd to the differential dµ̄(H) of a hyperplane H and the (d − 1)-dimensional volume
elements dλH (xj ) of points xj ∈ H, j = 1, . . . , d. Intuitively speaking, instead of choosing d random
points in Rd , we first choose a random hyperplane and then, in a second step, we choose d random points
in this hyperplane. More precisely, the corresponding transformation involves a Jacobian of the form
[x1 , . . . , xd ].
In this paper we need an analogous formula for two sets of points. The points x1 , . . . , x2d−k determine two hyperplanes H1 , H2 which are the affine span of x1 , . . . , xd and xd−k+1 , . . . , x2d−k , respectively. The following formula of Blaschke-Petkantschin type relates the d-dimensional volume elements
dλd (xj ), j = 1, . . . , 2d − k, to the differentials dµ̄(H1 ), dµ̄(H2 ) of the hyperplanes H1 , H2 , to the
(d − 1)-dimensional volume elements dλH1 (xj ) and dλH2 (xl ) of points xj , j = 1, . . . , d − k and xl ,
l = d + 1, . . . , 2d − k, which are contained in exactly one hyperplane, and to the (d − 2)-dimensional
volume elements dλH1 ∩H2 (xj ) of points xj , j = d − k + 1, . . . , d, which are contained in both hyperplanes. 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. Let 0 ≤ k ≤ d − 1, and let g : (Rd )2d−k → R be a nonnegative measurable function.
Then
Z
Z
···
Rd
g(x1 , . . . , x2d−k )
2d−k
Y
dλd (xj )
(3.1)
j=1
Rd
2
Z
Z
Z
Z
Z
···
= Γ(d)
d
d
H1
Hd−1
Hd−1
Z
Z
···
Z
···
H1 H1 ∩H2 H1 ∩H2 H2
g(x1 , . . . , x2d−k )
H2
× λH1 ([x1 , . . . , xd ])λH2 ([xd−k+1 , . . . , x2d−k ]) (sin ϕ)−k
×
2d−k
Y
j=d+1
dλH2 (xj )
d
Y
dλH1 ∩H2 (xj )
d−k
Y
j=1
j=d−k+1
where sin ϕ denotes the sinus of the angle between H1 and H2 .
6
dλH1 (xj ) dµ̄(H1 ) dµ̄(H2 ),
Proof. The Blaschke-Petkantschin formula applied to x1 , . . . , xd and Fubini’s theorem show that
Z
Z
···
Rd
g(x1 , . . . , x2d−k )
dλd (xj )
j=1
Rd
Z
Z
Z Z
···
= Γ(d)
H1
d
Hd−1
×
2d−k
Y
2d−k
Y
Z
···
H1 Rd
dλd (xj )
g(x1 , . . . , x2d−k )λH1 ([x1 , . . . , xd ])
Rd
d
Y
dλH1 (xj ) dµ̄(H1 ).
j=1
j=d+1
We fix H1 and set
Z
Z Z
I(f ) =
Z
···
H1
···
H1 Rd
f (xd−k+1 , . . . , x2d−k )
2d−k
Y
j=d+1
Rd
d
Y
dλd (xj )
dλH1 (xj )
j=d−k+1
for nonnegative measurable functions f : (Rd )d → R.
An essential ingredient of our proof is a special case of a generalized linear Blaschke-Petkantschin
formula due to Vedel Jensen and Kiêu [14] (see also [26, Theorem 5.6, p. 135]). For all nonnegative
measurable functions h and for a (fixed) hyperplane H, we thus have
Z
Z Z
···
H
Z
···
H Rd
Z
ZR
h(y1 , . . . yd−1 )
dλd (yj )
Z Z
···
Ldd−1 H∩L
k
Y
dλH (yj )
j=1
j=k+1
d
= Γ(d)
d−1
Y
Z
···
H∩L L
h(y1 , . . . yd−1 )
L
d−1
Y
× λL ([0, y1 , . . . , yd−1 ]) (sin ϕ)−k
dλL (yj )
j=k+1
k
Y
dλH∩L (yj ) dν̄(L),
j=1
where ϕ denotes the angle between H and L, and ν̄ is the rotation invariant Haar measure on the Grassmannian Ldd−1 of (d − 1)-dimensional linear subspaces in Rd with total measure dκd /2.
Using a standard argument (cf., e.g., Schneider and Weil [25], § 6.1) this immediately gives a generalized affine Blaschke-Petkantschin formula for I(f ). Put H = H1 − x2d−k and xd−k+j − x2d−k = yj ;
then
Z Z
Z Z
Z
···
· · · f (y1 + x2d−k , . . . , yd−1 + x2d−k , x2d−k )
I(f ) =
Rd H
×
H Rd
d−1
Y
Rd
dλd (yj )
Z
dλH (yj ) dλd (x2d−k )
j=1
j=k+1
Z
k
Y
Z
Z Z
···
=Γ(d)
Rd Ldd−1 H∩L
Z
···
H∩L L
f (y1 + x2d−k , . . . , yd−1 + x2d−k , x2d−k )
L
× λL ([0, y1 , . . . , yd−1 ]) (sin ϕ)−k
×
d−1
Y
j=k+1
dλL (yj )
k
Y
dλH∩L (yj ) dν̄(L) dλd (x2d−k )
j=1
7
Z
Z
Z
Z
···
=Γ(d)
d
H1 ∩H2
Hd−1
Z
···
H1 ∩H2 H2
f (xd−k+1 , . . . , x2d−k )
H2
× λH2 ([xd−k+1 , . . . , x2d−k ]) (sin ϕ)−k
×
2d−k
Y
d
Y
dλH2 (xj )
dλH1 ∩H2 (xj ) dµ̄(H2 ).
j=d−k+1
j=d+1
Setting f (xd−k+1 . . . , x2d−k ) = Γ(d)g(x1 , . . . , x2d−k )λH1 ([x1 , . . . , xd ]) for fixed x1 , . . . xd−k , one can
easily complete the proof of the lemma.
4
Some auxiliary estimates
A random point X in Rd is said to be normally distributed with positive definite d × d-covariance matrix
Σ and mean 0, i.e., X ∼ N (0, Σ), if it is chosen according to the density
1
fΣ (x) = p
(2π)d det Σ
1 T −1
Σ x
e− 2 x
,
x ∈ Rd .
For simplicity, we will exclusively consider the case Σ = 12 Id . In this case we put fΣ (x) = φd (x), or
simply fΣ (x) = φ(x) if d = 1. The one-dimensional normal distribution N (0, 21 ) is given by
Zz
Φ(z) :=
Zz
1
φ(x) dx = √
π
−∞
2
e−x dx,
z ∈ R.
−∞
The corresponding measures having these functions as densities with respect to the appropriate Lebesgue
measure, will be denoted by dφd (x) instead of φd (x)dx, etc.
We will repeatedly use the following well known asymptotic expansions concerning the density of
the normal distribution:
Lemma 4.1. Let j ≥ 0 and γ > 0. Then, as h1 → ∞,
Z∞
Γ(j + 1) −(j+1)
(h2 − h1 )j φ(h2 )γ dh2 =
h1
φ(h1 )γ 1 + O h−2
.
1
j+1
(2γ)
h1
Lemma 4.2. For n ∈ N, α, β ∈ R and γ > 0,
Z∞
Φ(h1 )n−α hβ1 φ(h1 )γ dh1 = Γ(γ)2γ−1 n−γ (ln n)
β+γ−1
2
(1 + o(1))
1
as n → ∞.
The proof of Lemma 4.1 is immediate by the substitution t = 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.
8
Lemma 4.3. Let j, l ≥ 0 and γ > 0. Then there exists a constant c > 0 depending only on j, l, γ such
that, for h1 ≥ 1,
Z∞ Zπ
h1 sin ϕ
−(j+l+2)
j
γ
(h2 − h1 ) φ(h2 ) φ
(sin ϕ)l dϕ dh2 ≤ c h1
φ(h1 )γ .
2
h1 0
Proof. Since sin is symmetric with respect to π2 , by Fubini’s theorem and Lemma 4.1, we get
Z∞ Zπ
j
γ
(h2 − h1 ) φ(h2 ) φ
h1 sin ϕ
2
(sin ϕ)l dϕ dh2
h1 0
π
Z∞
Z2 h1 sin ϕ
j
γ
(sin ϕ)l dϕ
≤ 2 (h2 − h1 ) φ(h2 ) dh2 φ
2
0
h1
π
2
≤
−(j+1)
2c1 h1
φ(h1 )γ
Z
φ
h1 ϕ
π
ϕl dϕ,
0
where π2 ϕ ≤ sin ϕ ≤ ϕ for 0 ≤ ϕ ≤
j, γ. Substituting h1 ϕ = t, we obtain
π
2
was used in the last step. Here the constant c1 depends only on
π
Z2
φ
h1 ϕ
π
l
ϕ dϕ ≤
Z∞ t l
−(l+1)
φ
t dt ≤ c2 h1
,
π
−(l+1)
h1
0
0
where c2 is a constant depending only on l. Thus the assertion follows.
Lemma 4.4. Let j, l ≥ 0 and γ > 0. Then there exists a constant c > 0 depending only on j, l, γ such
that, for h1 ≥ 1,
Z∞ Zπ
h1 − h2 cos ϕ
−(j+l+1)
j
γ
(h2 − h1 ) φ(h2 ) φ
(sin ϕ)l−1 dϕ dh2 ≤ c h1
φ(h1 )γ .
2 sin ϕ
h1 0
Proof. We will use Fubini’s theorem repeatedly and let c1 , c2 , . . . denote constants depending only on
j, l, γ. Then, first substituting (for fixed h2 )
q
h2 − h1 cos ϕ
h1 − h2 cos ϕ
u=
, du =
dϕ
=
u2 + h22 − h21 sin−1 ϕ dϕ
sin ϕ
sin2 ϕ
with
1
sin ϕ = 2
u + h22
and then h2 = h1 +
Z∞ Zπ
s2
2h1 ,
h1 u + h2
q
u2
+
h22
−
h21
,
we get
(h2 − h1 )j φ(h2 )γ φ
h1 − h2 cos ϕ
2 sin ϕ
(sin ϕ)l−1 dϕ dh2
h1 0
Z∞ Z∞
=
h1 −∞
l
p
u h1 u + h2 u2 + h22 − h21
p
(h2 − h1 )j φ(h2 )γ φ
du dh2
2
u2 + h22 − h21 (u2 + h22 )l
9
≤
γ
c1 h−j
1 φ(h1 )
Z∞ Z∞
γ
φ(s) φ
−∞ 0
s2
2h1
γ
φ
u
2
s2j h−2l
1
q
s2
h1 u + h1 + 2h
u2 + s2 +
1
q
×
s4
u2 + s2 + 4h
2
s4
4h21
l
s
ds du
h1
1
−(j+l+1)
≤ c2 h1
φ(h1 )γ
Z∞ Z∞
φ(s)γ φ
−∞ 0
×
≤
−(j+l+1)
c2 h1
φ(h1 )γ
Z∞ Z∞
φ(s)γ φ
u
2
s2j q
s
4
s
u2 + s2 + 4h
2
1
s
!l
4
s2
s
|u| + 1 + 2
u2 + s2 + 2
ds du
2h1
4h1
u h
2
l i
s2j |u| + (1 + s2 )(|u| + s + s2 )
ds du
−∞ 0
−(j+l+1)
≤ c h1
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].
If S = S(Y1 , . . . , Yn ) is any real symmetric function of the independent identically distributed
ran1 Pn+1
dom vectors Yj , 1 ≤ j < ∞, we set Si = S(Y1 , . . . , Yi−1 , Yi+1 , . . . , Yn+1 ) and S(.) = n+1
i=1 Si .
The Efron-Stein jackknife inequality then says
Var S ≤ E
n+1
X
(Si − S(.) )2 = (n + 1)E (Sn+1 − S(.) )2 .
(5.1)
i=1
Note that the right-hand side is not decreased if S(.) is replaced by any other function of Y1 , . . . , Yn+1 .
We apply this inequality to the random variable f (Pn ) where f (·) is a measurable function of convex
polytopes. Then S = f ([X1 , . . . , Xn ]) = f (Pn ), and we replace S(.) by f (Pn+1 ) which is a function
of the convex hull of Pn and a further random point Xn+1 . The Efron-Stein jackknife inequality then
yields that
Var f (Pn ) ≤ (n + 1)E (f (Pn ) − f (Pn+1 ))2 .
(5.2)
In the case that f (·) is the number of i-faces of Pn , we obtain
Var fi (Pn ) ≤ (n + 1)E (fi (Pn+1 ) − fi (Pn ))2 .
Let Pn be fixed and choose the additional random point Xn+1 . If the point Xn+1 is contained in Pn ,
the random variable fi (Pn+1 ) − fi (Pn ) equals 0. If Xn+1 ∈
/ Pn , the relative interior of some of the
i-dimensional faces of Pn is contained in the interior of [Pn , Xn+1 ], let fi− (Xn+1 ) be the number of
theses faces, and some of the i-dimensional faces of [Pn , Xn+1 ] are not contained in Pn , let fi+ (Xn+1 )
be the number of those faces. Then we have
|fi ([Pn , Xn+1 ]) − fi (Pn )| = fi+ (Xn+1 ) − fi− (Xn+1 ) ≤ fi+ (Xn+1 ) + fi− (Xn+1 ) .
10
Since Pn is simplicial with probability one, this number can easily be estimated in terms of the number
Fn (Xn+1 ) of facets of Pn which can be seen from Xn+1 . Here Fn (Xn+1 ) = 0 if Xn+1 is contained in
Pn , and if Xn+1 ∈
/ Pn then Fn (Xn+1 ) > 0 is the number of facets of Pn which are – up to (d − 2)dimensional faces – contained in the interior of the convex hull of Pn and Xn+1 . Now each i-dimensional
“new” face of [Pn , Xn+1 ] not contained in Pn is the convex hull of Xn+1 and an (i − 1)-dimensional
face of Pn . Since this (i − 1)-dimensional face is also a face of a facet of Pn which can be seen from
Xn+1 , and each facet is a simplex, we obtain
d
fi+ (Xn+1 ) ≤
Fn (Xn+1 ) .
i
On the other hand each i-dimensional face of Pn which is – up to (i − 1)-dimensional faces – contained
in the interior of [Pn , Xn+1 ] is also a face of a facet contained in the interior of [Pn , Xn+1 ]. Hence
d
fi− (Xn+1 ) ≤
Fn (Xn+1 )
i+1
and combining these estimates proves
2
E(fi (Pn+1 ) − fi (Pn )) ≤
d+1
i+1
2
EFn (Xn+1 )2 .
(5.3)
Thus each estimate for the second moment of Fn (Xn+1 ) yields an estimate for Var fi (Pn ), and hence
Theorem 1.1 follows from Theorem 1.5.
6
6.1
Proof of Theorem 1.5
Asymptotic expansion of the expectation
We start with the proof of the first part of Theorem 1.5. The case d = 1 is included by proper interpretations of the subsequent arguments. Choose n + 1 independent normally distributed random points
X1 , . . . Xn , X in Rd . The convex hull of the first n points is a Gaussian polytope Pn , and with probability one Pn is simplicial. For I ⊂ {1, . . . , n} with |I| = d, denote by FI the convex hull of {Xi : i ∈ I}
which is a (d − 1)-dimensional simplex. The affine hull of FI is denoted by H(FI ). With probability
one, this affine hull is a hyperplane which dissects Rd into two (closed) halfspaces. The halfspace which
contains the origin will be denoted by H0 (FI ), the other by H+ (FI ). The origin is contained in exactly
one halfspace with probability one. In the following, we want to assume that Pn contains the origin. This
happens with high probability, since by Wendel’s theorem [28]
P(0 ∈
/ Pn ) = O(nd 2−n ),
an thus the condition that Pn contains the origin can be inserted by adding a suitable error term. Moreover, we can also assume that the points X1 , . . . , Xn , X are in general relative position (i.e. any subset
of at most d + 1 of these random points is affinely independent).
We are interested in the number of facets of Pn which can be seen from the additional random point
X∈
/ Pn . Denote the set of these facets by Fn (X), i.e.,
Fn (X) = F(X1 , . . . , Xn ; X)
= {FI : Pn ⊂ H0 (FI ), X ∈ H+ (FI ), I ⊂ {1, . . . , n}, |I| = d} .
Here we can define Fn (X) as the empty set, if the origin is not contained in the interior of Pn or if
the random points are not in general relative position. Similar definitions can be given for deterministic
11
points x1 , . . . , xn and x in general relative position such that the convex hull of x1 , . . . , xn contains
the origin. When applying Wendel’s theorem in considering EFn (X), the error
term is of the order
n
2d
−n
−n
n 2 = O(c ) with a suitable constant c > 1, since Fn (X) is bounded by d . Using this we have
Z
···
EFn (X) =
Z X
1{FI ∈ Fn (x)}
dφd (xj ) dφd (x) + O(c−n ),
(6.1)
j=1
Rd
Rd
n
Y
where the summation extends over all subsets I ⊂ {1, . . . , n} with |I| = d. Denote by F1 the convex
hull of x1 , . . . , xd . Then
Z
Z
n
Y
n
· · · 1{F1 ∈ Fn (x)}
dφd (xj ) dφd (x) + O(c−n ).
EFn (X) =
d
Rd
j=1
Rd
The probability content of the halfspace H+ (F1 ) is
Z
dφd (x) = 1 − Φ(h1 ),
H+ (F1 )
where h1 is the distance of H(F1 ) to the origin. If F1 ∈ Fn (X), then X is contained in the halfspace
H+ (F1 ) with probability content 1−Φ(h1 ), and the random points Xj , j ∈ {d+1, . . . , n}, are contained
in the halfspace H0 (F1 ) with probability content Φ(h1 ). Hence we obtain
Z
Z
d
Y
n
EFn (X) =
· · · Φ(h1 )n−d (1 − Φ(h1 ))
dφd (xj ) + O(c−n ).
d
Rd
j=1
Rd
Parameterizing the hyperplane H(F1 ) =: H1 by its distance h1 ≥ 0 from the origin and its unit normal
vector u1 in the form H1 = H(u1 , h1 ), and using the affine Blaschke-Petkantschin formula, we find that
Z Z∞
n
EFn (X) =Γ(d)
Φ(h1 )n−d (1 − Φ(h1 ))
d
d−1 0
S

d
Z Z

Y
×
· · · λH1 ([x1 , . . . , xd ])
(φd (xj ) dλH1 (xj )) dh1 dσ(u1 ) + O(c−n ).


H1
j=1
H1
(d−1)
The inner integral (in brackets) is the expected volume EVd−1 (Pd
Gaussian simplex in Rd−1 times φ(h1 )d , which gives
) of a random (d − 1)-dimensional
Z Z∞
n
(d−1)
EFn (X) = Γ(d)
EVd−1 (Pd
)
Φ(h1 )n−d (1 − Φ(h1 )) φ(h1 )d dh1 dσ(u1 ) + O(c−n ).
d
Sd−1 0
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
(d−1)
EFn (X) = 2d−1 κd Γ(d + 1)EVd−1 (Pd
This proves the first part of Theorem 1.5.
12
) n−1 (ln n)
d−1
2
(1 + o(1)).
(6.2)
6.2
Estimate of the variance
The main part of the proof is devoted to estimating the second moment of Fn (X). In the case d = 1, we
have Fn (X) = Fn (X)2 , hence the assertion follows. Now let d ≥ 2.
As in (6.1) we have
!2 n
Z
Z X
Y
2
EFn (X) = · · ·
1{FI ∈ Fn (x)}
dφd (xj ) dφd (x) + O(c−n )
Rd
j=1
I
Rd
with some c > 1. The summation extends over all subsets I ⊂ {1, . . . , n} with |I| = d. We expand the
integrand and get
Z
n
Y
XXZ
· · · 1{FI , FJ ∈ Fn (x)}
dφd (xj ) dφd (x) + O(c−n ),
(6.3)
EFn (X)2 =
I
J
Rd
j=1
Rd
where the summation extends over all subsets I, J ⊂ {1, . . . , n} with |I| = |J| = d. If we fix the
number k = |I ∩ J| ∈ {0, . . . , d}, then the corresponding term in (6.3) depends only on k and not on
(k)
the particular choice of I and J. For given k ∈ {0, . . . , d}, we put F1 = [X1 , . . . , Xd ] and F2 =
(d)
[Xd−k+1 , . . . , X2d−k ]. Note that for k = d we have F1 = F2 . Hence EFn (X)2 can be rewritten as
Z
Z
d n
X
Y
n d n−d
(k)
2
EFn (X) =
· · · 1{F1 , F2 ∈ Fn (x)}
dφd (xj ) dφd (x) + O(c−n ).
d
k
d−k
k=0
Rd
j=1
Rd
The summand corresponding to k = d is just EFn (X), and thus (6.2) yields
Z
Z
d−1
n
Y
X
(k)
2
2d−k
· · · 1{F1 , F2 ∈ Fn (x)}
dφd (xj ) dφd (x)
EFn (X) ≤ c1
n
k=0
Rd
j=1
Rd
d−1
+ O n−1 (ln n)
2
;
here and in the following c1 , c2 , . . . denote constants which are independent of n. The summand corresponding to k = d also yields the asserted lower bound for EFn (X)2 .
(k)
Let h1 , h2 be the distance to the origin and u1 , u2 the unit normal vector of H(F1 ), H(F2 ), respec(k)
tively, such that H(F1 ) = H(u1 , h1 ) and H(F2 ) = H(u2 , h2 ). Since the integrand is symmetric in F1
(k)
and F2 , we restrict our integration to h1 ≤ h2 . Thus we get
Z
Z
d−1
n
X
Y
(k)
2
2d−k
EFn (X) ≤ c2
n
· · · 1{h1 ≤ h2 } 1{F1 , F2 ∈ Fn (x)}
dφd (xj ) dφd (x)
k=0
Rd
j=1
Rd
+ O n−1 (ln n)
d−1
2
.
(k)
(k)
If F1 , F2 ∈ Fn (x), then the points x2d−k+1 , . . . , xn are contained in H0 (F1 ) ∩ H0 (F2 ), and the
(k)
corresponding measure of the set of these points is at most Φ(h1 )n−2d+k . Moreover, F1 , F2 ∈ Fn (x)
(k)
(k)
implies that x is contained in H+ (F1 ) ∩ H+ (F2 ). Denote the distance of H+ (F1 ) ∩ H+ (F2 ) to the
origin by h12 . Then the corresponding measure is at most 1 − Φ(h12 ). This yields
Z
Z
d−1
2d−k
Y
X
EFn (X)2 ≤ c2
n2d−k
· · · 1{h1 ≤ h2 } Φ(h1 )n−2d+k (1 − Φ(h12 ))
dφd (xj )
k=0
Rd
j=1
Rd
+ O n−1 (ln n)
d−1
2
.
13
The range of integration can be reduced further to 1{1 ≤ h1 ≤ h2 }, since the additional error term is of
the order n2d Φ(1)n−2d+k = O(c−n ) with a suitable c > 1. For h1 ≥ 1 and since h12 ≥ h1 , Lemma 4.1
implies that 1 − Φ(h12 ) ≤ c3 h−1
1 φ(h12 ). Therefore we can conclude that
EFn (X)2 ≤ c4
d−1
X
Z
Z
n2d−k
···
k=0
1{1 ≤ h1 ≤ h2 } Φ(h1 )n−2d+k h−1
1 φ(h12 )
2d−k
Y
j=1
Rd
Rd
+ O n−1 (ln n)
dφd (xj )
d−1
2
.
(6.4)
To develop an estimate for the integral
Z
Z
Ik =
1{1 ≤ h1 ≤ h2 } Φ(h1 )n−2d+k h−1
1 φ(h12 )
···
2d−k
Y
Rd
Rd
dφd (xj )
j=1
in the cases where 0 ≤ k ≤ d − 1, we apply the Blaschke-Petkantschin formula (3.1) and obtain
Z Z∞ Z Z∞
2
Ik =Γ(d)
−k
Φ(h1 )n−2d+k h−1
1 φ(h12 ) (sin ϕ)
Sd−1 1 Sd−1 h1
× Jk (u1 , h1 , u2 , h2 ) dh2 dσ(u2 ) dh1 dσ(u1 )
(6.5)
with
Jk (u1 , h1 , u2 , h2 ) =

Z
Z
Z
···

Z
Z
Z
···
···
H1 H1 ∩H2 H1 ∩H2 H2
H1
×
2d−k
Y
(k)
λH1 (F1 )λH2 (F2 )
dλH2 (xj )
j=d+1
φd (xj )
j=1
H2
d
Y
2d−k
Y
dλH1 ∩H2 (xj )
d−k
Y
dλH1 (xj )
.
(6.6)

j=1
j=d−k+1


In the following, we distinguish the cases k = 0 and k ∈ {1, . . . , d − 1}.
(d−2)
First, let k ∈ {1, . . . , d − 1}. We denote by h12
the distance of the origin to the (d − 2)-dimensional
(d−2)
∗
intersection H1 ∩ H2 and define an angle ϕ ∈ (0, π/2) by h1 = h2 cos ϕ∗ . Then h12 = h2 ≤ h12
if
(d−2)
∗
∗
0 ≤ ϕ ≤ ϕ , and h12 = h12
if ϕ ≥ ϕ . By v1 and v2 we denote unit vectors parallel to H1 and H2 ,
(k)
respectively, orthogonal to H1 ∩ H2 . The (d − 1)-volumes of the simplices F1 and F2 can be bounded
(k)
from above by the (d − 2)-volumes of their projections to H1 ∩ H2 , projH1 ∩H2 F1 and projH1 ∩H2 F2 ,
and their heights in direction orthogonal to H1 ∩ H2 . Hence we get
!
r
λH1 (F1 ) ≤ λH1 ∩H2 (projH1 ∩H2 F1 )
max
j=1,...,d−k
|hxj , v1 i| +
(d−2) 2
h12
where the height of F1 was estimated by the distance of H1 ∩H2 to h1 u1 , which is
− h21
(d−2) 2
h12
−h21
and the maximal distance of the points xj to the point h1 u1 in direction v1 . Analogously,
r
(k)
(k)
λH2 (F2 ) ≤ λH1 ∩H2 (projH1 ∩H2 F2 )
max
j=d+1,...,2d−k
14
|hxj , v2 i| +
,
(d−2) 2
h12
− h22
!
.
1/2
,
Writing xj ∈ H1 , j = 1, . . . d − k, as the orthogonal sum xj = h1 u1 + x1j v1 + yj , where yj is contained
in the (d − 2)-dimensional linear subspace parallel to H1 ∩ H2 , which we identify with Rd−2 , we have
hxj , v1 i = x1j and φd (xj ) = φ(h1 )φ(x1j )φd−2 (y). For the integration with respect x1j , we obtain
Z
Z
···
max
j=1,...,d−k
|x1j |
(φd (xj ) dx1j ) ≤ c5
j=1
R
R
d−k
Y
d−k
Y
(φ(h1 )φd−2 (yj )),
j=1
and an analogous result holds for xj ∈ H2 , j = d + 1, . . . , 2d − k. Recall that h1 ≤ h2 . This shows that
Jk (u1 , h1 , u2 , h2 ) ≤
c5 +
×
r
!2
(d−2) 2
h12

Z
−
h21
(d−2) k
φ(h1 )d−k φ(h2 )d−k φ(h12
)
Z
···
λd−2 ([y1 , . . . , yd ])λd−2 ([yd−k+1 , . . . y2d−k ])

Rd−2 Rd−2
2d−k
Y
(φd−2 (yj ) dλd−2 (yj ))


.
(6.7)

j=1
Using that (a + b)2 ≤ 2(a2 + b2 ) and (1 + x/2)e−x/2 ≤ 1, for a, b ∈ R and x ≥ 0, we deduce that
d−k+ 12
Jk (u1 , h1 , u2 , h2 ) ≤ c6 φ(h1 )
d−k
φ(h2 )
φ
1
(d−2) k− 2
h12
,
(6.8)
and thus
Z Z∞ Z Z∞
Ik ≤ c7
1
d−k+ 2
Φ(h1 )n−2d+k h−1
φ(h2 )d−k
1 φ(h1 )
Sd−1 1 Sd−1 h1
1
(d−2) k− 2
× φ(h12 )φ h12
(sin ϕ)−k dh2 dσ(u2 ) dh1 dσ(u1 ).
Since the integrand is rotation invariant, we may assume that u1 = ed , and hence arrive at
Z∞ Z Z∞
Ik ≤ c8
1
d−k+ 2
Φ(h1 )n−2d+k h−1
φ(h2 )d−k
1 φ(h1 )
1 Sd−1 h1
× φ(h12 )φ
1
(d−2) k− 2
h12
(sin ϕ)−k dh2 dσ(u2 ) dh1 .
(6.9)
Elementary calculations show that
(d−2) 2
h12
= h21 +
(h2 − h1 cos ϕ)2
(h1 − h2 cos ϕ)2
2
=
h
+
,
2
(sin ϕ)2
(sin ϕ)2
moreover,
(d−2) 2
h12
≥
h212
=
h22
≥
≥
h21
and
h212
=
(d−2) 2
h12
h21
(sin ϕ)2
1+
4
(sin ϕ)2
1+
4
15
,
,
ϕ ∈ (0, π);
ϕ ∈ (0, ϕ∗ ],
ϕ ∈ [ϕ∗ , π).
(6.10)
(6.11)
(6.12)
We parameterize u2 ∈ Sd−1 \ {ed , −ed } by the angle ϕ enclosed by u2 and ed and by its normalized
projection v to Rd−1 , i.e. v ∈ Sd−1 ∩e⊥
d and ϕ ∈ (0, π). Let σd−2 denote the spherical Lebesgue measure
d−2
on S .
Then we can estimate the inner double integral in (6.9) by using that k −
Z∞ Z
1
2
≥
1
4
and Lemma 4.4
1
(d−2) k− 2
(sin ϕ)−k dσ(u2 ) dh2
φ(h2 )d−k φ(h12 )φ h12
h1 Sd−1
k+ 21
Z∞ Z
Zπ
≤ c9 φ(h1 )
d−k
φ(h2 )
S d−2
h1
k+ 12
φ
h1 − h2 cos ϕ
2 sin ϕ
(sin ϕ)d−k−2 dϕ σd−2 (v) dh2
0
Z∞ Zπ
≤ c10 φ(h1 )
d−k
φ(h2 )
φ
h1 − h2 cos ϕ
2 sin ϕ
(sin ϕ)d−k−2 dϕ dh2
h1 0
≤
1
−(d−k)
c11 h1
φ(h1 )d+ 2 .
(6.13)
Then, for k ∈ {1, . . . , d − 1}, we deduce from (6.9) and (6.13) that
Z∞
Ik ≤ c12
Φ(h1 )n−2d+k h−d+k−1
φ(h1 )2d+1−k dh1 ≤ c13 n−(2d−k+1) (ln n)
1
d−1
2
.
(6.14)
1
Finally, we consider the case k = 0. Using the previous notation and (6.11) and (6.12), we get directly
from (6.5) and (6.6)
Z Z∞ Z Z∞
I0 ≤ c14
Φ(h1 )n−2d h1 −1 φ(h12 )φ(h1 )d φ(h2 )d dh2 dσ(u2 ) dh1 dσ(u1 )
Sd−1 1 Sd−1 h1
Z∞
n−2d
≤ c15
Φ(h1 )
h1
−1
d+1
Z∞ Zπ
φ(h1 )
1
d
φ(h2 ) φ
h1 sin ϕ
2
(sin ϕ)d−2 dϕ.
h1 0
By Lemma 4.3, we obtain
Z∞
I0 ≤ c16
Φ(h1 )n−2d h1 −(d+1) φ(h1 )2d+1 dh1 ≤ c17 n−(2d+1) (ln n)
d−1
2
.
(6.15)
1
Combining the estimates (6.4), (6.14) and (6.15), we arrive at
EFn (X)2 ≤ c18
d
X
n2d−k n−2d+k−1 (ln n)
d−1
2
≤ c19 n−1 (ln n)
d−1
2
.
k=0
This completes the proof of Theorem 1.5.
7
Proof of Theorem 1.3
The proof is based on arguments similar to those involved in the proofs of Theorems 1.1 and 1.5. Therefore we use the same notation as before. In particular, c1 , c2 , . . . denote constants which merely depend
on the dimension.
16
Denote by νi the Haar probability measure on the set Ldi of i-dimensional linear subspaces in Rd .
Kubota’s theorem and the rotation invariance of the normal distribution immediately give
!
Z
EVi (Pn ) = E cd,i
Ldi
Vi (projL Pn ) dνi (L)
= cd,i EVi (projL0 Pn )
with cd,i = di κd / (κi κd−i ), for an arbitrary i-dimensional linear subspace L0 which we identify with
Ri . The projection to Ri of a Gaussian sample in Rd again is a Gaussian sample. Hence
EVi (Pn ) = cd,i EVi (Pn(i) ),
(i)
where Pn is the convex hull of a Gaussian sample in Ri . For the variance we obtain by Hölder’s
inequality
Var Vi (Pn ) = E(Vi (Pn ) − EVi (Pn ))2
!2
Z Vi (projL Pn ) − EVi (Pn(i) ) dνi (L)
= E cd,i
Ldi
≤
Z
c2d,i E
Ldi
Vi (projL Pn ) −
2
EVi (Pn(i) )
!
dνi (L) .
Thus Fubini’s theorem and the rotation invariance imply
2
Var Vi (Pn ) ≤ c2d,i E Vi (Pn(i) ) − EVi (Pn(i) ) = c2d,i Var Vi (Pn(i) ).
(7.1)
Therefore by the Efron-Stein jackknife inequality (5.2) it suffices to prove
Var Vd (Pn ) ≤ (n + 1)E (Vd (Pn+1 ) − Vd (Pn ))2 ≤ c1 (ln n)
d−3
2
(7.2)
(i.e., the case i = d), which then yields the general result.
√
We will assume that Pn contains the origin, which leads to an error term of the order n5d 2−n =
O(c−n ) with a suitable constant c > 1. To see this, one can use Cauchy’s inequality
p
E (Vd (Pn+1 ) − Vd (Pn ))2 1{0 ∈
/ Pn } ≤ E ((Vd (Pn+1 ) − Vd (Pn ))4 ) P(0 ∈
/ Pn )
and Hölder’s inequality
!4
4
4
E (Vd (Pn+1 ) − Vd (Pn )) ≤ E Vd (Pn+1 ) ≤ E
X
Vd ([0, FI ])
≤
I
n+1
d
4
EVd ([0, F1 ])4 ;
here the summation extends over all sets I ⊂ {1, . . . , n + 1} with |I| = d. Hence we obtain
!2
X
E (Vd (Pn+1 ) − Vd (Pn ))2 = E
1{FI ∈ Fn (X)} Vd ([FI , X]) + O(c−n )
=2
I
d
X
k=0
×
×
n
d
Z
Z
d n−d
· · · 1{1 ≤ h1 ≤ h2 }
k
d−k
Rd
(k)
1{F1 , F2
n
Y
Rd
dφd (xj ) dφd (x) + O(c−n )
j=1
17
(k)
∈ Fn (x)}Vd ([F1 , x])Vd ([F2 , x])
≤ c2
d
X
n
k=0
×
2d−k
Z
Z
···
Rd
(k)
1{1 ≤ h1 ≤ h2 } 1{F1 , F2
∈ Fn (x)}
Rd
(k)
Vd ([F1 , x])Vd ([F2 , x])
n
Y
dφd (xj ) dφd (x) + O(c−n ).
j=1
For the integration with respect to x, observe that Vd ([F1 , x]) equals d1 λH1 (F1 ) – which is independent
(k)
of x – times the distance hx, u1 i − h1 of x to H1 = H(F1 ), and a similar assertion holds for F2 .
Let us first consider the summand corresponding to k = d. In this case, we can estimate
n
d
Z
Z
1{1 ≤ h1 } 1{F1 ∈ Fn (x)}Vd ([F1 , x])2
···
Rd
≤ n
dφd (xj ) dφd (x)
j=1
Rd
d
n
Y
Z
Z
Φ(h1 )n−d 1{1 ≤ h1 }
···
Rd
Rd
Z
1{x ∈ H + (F1 )} (hx, u1 i − h1 )2 dφd (x)λH1 (F1 )2
×
Z
Z
Φ(h1 )n−d 1{1 ≤ h1 }
···
Rd
Rd
≤ c3 nd
Z
≤ c4 nd
∞
1
≤ c5 n
−1
(h − h1 )2 φ(h) dh λH1 (F1 )2
d
Y
dφd (xj )
j=1
2
Φ(h1 )n−d 1{1 ≤ h1 }h−3
1 φ(h1 )λH1 (F1 )
···
Z
Z∞
h1
Z
Rd
dφd (xj )
j=1
Rd
= nd
d
Y
d
Y
dφd (xj )
j=1
Rd
d+1
dh1
Φ(h1 )n−d h−3
1 φ(h1 )
(ln n)
d−3
2
.
where the affine Blaschke-Petkantschin formula and Lemma 4.2 were applied in the fourth step.
If d = 1, then only the case k = 1 can occur, and the proof is finished at this point. Subsequently,
we consider the case d ≥ 2. In order to estimate the summands corresponding to 0 ≤ k ≤ d − 1,
we need some preparations. Denote again by ϕ the angle between u1 and u2 , by h12 the distance of
(k)
H+ (F1 ) ∩ H+ (F2 ) to the origin, and by u12 the unit vector pointing from the origin to the nearest
(k)
(k)
point of H+ (F1 ) ∩ H+ (F2 ). Then H+ (F1 ) ∩ H+ (F2 ) is contained in {y ∈ Rd : hy, u12 i ≥ h12 }.
We use the parameterization x = hu12 + z with z ∈ u⊥
12 , where h ≥ h12 , and have
hx, ui i − hi = hhu12 , ui i + hz, ui i − hi ≤ h − h12 + kzk sin ϕ + h12 hu12 , ui i − hi
for ϕ ≤
ϕ≤
π
2,
π
2,
since the angle between ui and u12 is bounded by the angle between u1 and u2 . If ϕ∗ ≤
(d−2)
then h12 = h12
(k)
and h12 u12 ∈ H(F1 ) ∩ H(F2 ), hence h12 hu12 , ui i = hi for i = 1, 2. If
(k)
0 < ϕ < ϕ∗ , then h12 u12 ∈ H(F2 ), h12 = h2 , u12 = u2 , and h12 hu12 , u1 i − h1 ≤ h2 − h1 . Hence,
in any case we have
Z n
o
(k)
1 x ∈ H+ (F1 ) ∩ H+ (F2 ) (hx, u1 i − h1 )(hx, u2 i − h2 ) dφd (x)
Rd
18
Z∞ Z
≤
(h − h12 + kzk sin ϕ + h2 − h1 )(h − h12 + kzk sin ϕ) dφd−1 (z) dφ(h)
h12 u⊥
12
−2
−1
2
≤ c6 h−3
12 + h12 (sin ϕ + (h2 − h1 )) + h12 (sin ϕ + (h2 − h1 ) sin ϕ) φ(h12 )
=: S1 (h1 , h2 , ϕ)
√
(d−2)
for ϕ ≤ π2 , where we used Lemma 4.1. For ϕ ≥ π2 , (6.10) implies that h12
= h12 ≥ 2h1 . Thus, for
ϕ ≥ π/2 we obtain
Z n
o
(k)
1 x ∈ H+ (F1 ) ∩ H+ (F2 ) (hx, u1 i − h1 )(hx, u2 i − h2 ) dφd (x)
Rd
Z∞
(h12 − h1 )2
≤ (h − h1 )2 φ(h) dh ≤ 3φ(h12 ) +
φ(h12 )
h12
h12
≤ c7 φ(ηh1 ) =: S2 (h1 , h2 , ϕ),
√
where η := (1 + 2)/2. In addition, we define S1 (h1 , h2 , ϕ) = 0 for ϕ > π/2 and S2 (h1 , h2 , ϕ) = 0
for ϕ < π/2, and then we put S(h1 , h2 , ϕ) := S1 (h1 , h2 , ϕ) + S2 (h1 , h2 , ϕ).
(k)
The probability that the random points X2d−k+1 , . . . , Xn are contained in H0 (F1 ) ∩ H0 (F2 ) can
be bounded from above by Φ(h1 )n−2d+k . Hence we get
E (Vd (Pn+1 ) − Vd (Pn ))2 ≤ c8
d−1
X
Z
n2d−k
k=0
Z
1{1 ≤ h1 ≤ h2 }Φ(h1 )n−2d+k S(h1 , h2 , ϕ)
···
Rd
Rd
2d−k
Y
(k)
× λH1 (F1 )λH2 (F2 )
d−3
dφd (xj ) + O n−1 (ln n) 2 .
(7.3)
j=1
For k ∈ {0, . . . , d − 1}, we apply the Blaschke-Petkantschin formula (3.1) to the integral
Z
Ik =
Z
···
Rd
(k)
1{1 ≤ h1 ≤ h2 }Φ(h1 )n−2d+k S(h1 , h2 , ϕ) λH1 (F1 )λH2 (F2 )
2d−k
Y
dφd (xj ).
j=1
Rd
Thus we obtain
2
Z Z∞ Z Z∞
Ik = Γ(d)
Φ(h1 )n−2d+k S(h1 , h2 , ϕ) (sin ϕ)−k
Sd−1 1 Sd−1 h1
× Jk (u1 , h1 , u2 , h2 ) dh2 dσ(u2 ) dh1 dσ(u1 )
with
Jk (u1 , h1 , u2 , h2 ) =

Z

H1
×
Z
···
Z
Z
Z
···
Z
···
H1 H1 ∩H2 H1 ∩H2 H2
2d−k
Y
j=d+1
dλH2 (xj )
d
Y
j=d−k+1
19
(k)
λH1 (F1 )2 λH2 (F2 )2
j=1
H2
dλH1 ∩H2 (xj )
2d−k
Y
d−k
Y
j=1
dλH1 (xj )



.
φd (xj )
(7.4)
The two cases k = 0 and k ∈ {1, . . . , d − 1} will be treated separately. We decompose Ik in the form
Ik = Ik1 + Ik2 corresponding to the decomposition S = S1 + S2 .
We start with the case k = 0 and obtain
I01
Z Z∞ Z Z∞
= c9
Φ(h1 )n−2d S1 (h1 , h2 , ϕ)
Sd−1 1 Sd−1 h1
Z
×
Z
H2
Z
···
H1
2d
Y
(0)
λH2 (F2 )2
···
Z
φd (xj )
j=d+1
H2
Z Z∞ Z Z∞
≤ c10
λH1 (F1 )2
d
Y
φd (xj )
j=1
H1
2d
Y
d
Y
dλH1 (xj )
j=1
dλH2 (xj )dh2 dσ(u2 ) dh1 dσ(u1 )
j=d+1
Φ(h1 )n−2d S1 (h1 , h2 , ϕ)φ(h1 )d φ(h2 )d dh2 dσ(u2 ) dh1 dσ(u1 )
Sd−1 1 Sd−1 h1
Z∞
≤ c11
π
Φ(h1 )n−2d φ(h1 )d
Z2 Z∞
1
S1 (h1 , h2 , ϕ)(sin ϕ)d−2 φ(h2 )d dh2 dϕ dh1 .
0 h1
Now we have to consider the five different summands into which S1 (h1 , h2 , ϕ) naturally decomposes
and estimate the corresponding integrals. Using that h12 ≥ h1 and applying repeatedly Lemma 4.3, we
thus obtain
π
Z∞ Z2
−(d+3)
S1 (h1 , h2 , ϕ) (sin ϕ)d−2 φ(h2 )d dh2 dϕ ≤ c12 h1
φ(h1 )d+1 .
h1 0
Hence we arrive at
I01
∞
Z
−(d+3)
Φ(h1 )n−2d h1
≤ c13
1
φ(h1 )2d+1 dh1 ≤ c14 n−(2d+1) (ln n)
d−3
2
.
Moreover,
I02
Z Z∞ Z Z∞
≤ c15
Φ(h1 )n−2d S2 (h1 , h2 , ϕ)φ(h1 )d φ(h2 )d dh2 dσ(u2 ) dh1 dσ(u1 )
Sd−1 1 Sd−1 h1
Z∞
n−2d
≤ c16
Φ(h1 )
d
φ(h1 )
π
2
1
Z∞
≤ c17
Zπ Z∞
φ(ηh1 )(sin ϕ)d−2 φ(h2 )d dh2 dϕ dh1
h1
2
2d+η
Φ(h1 )n−2d h−1
dh1
1 φ(h1 )
1
2
≤ c18 n−(2d+η ) (ln n)
≤ c19 n−(2d+1) (ln n)
2d+η 2 −1−1
2
d−3
2
,
as 1 < η 2 < 2. Thus we have
I0 ≤ c20 n−(2d+1) (ln n)
d−3
2
.
(7.5)
Next we consider the cases k ∈ {1, . . . , d − 1}. First, we obtain
1
(d−2) k− 1
2
Jk (u1 , h1 , u2 , h2 ) ≤ c21 φ(h1 )d−k+ 2 φ(h2 )d−k φ(h12
20
)
by arguing in a similar way as for proving (6.8). Thus we have to bound from above
Z∞ Z Z∞
1
(d−2) k− 1
2
Φ(h1 )n−2d+k S(h1 , h2 , ϕ)(sin ϕ)−k φ(h1 )d−k+ 2 φ(h2 )d−k φ(h12
)
dh2 dσ(u2 ) dh1 ,
1 Sd−1 h1
where ϕ is the angle between e1 and u2 . Parameterizing the unit sphere as before, we see that this integral
can be estimated from above by
Z∞ Z∞ Z∞
(d−2) k− 1
2
1
Φ(h1 )n−2d+k φ(h1 )d−k+ 2 S(h1 , h2 , ϕ)φ(h2 )d−k φ(h12
)
(sin ϕ)d−k−2 dϕ dh2 dh1
1 h1 0
up to a constant multiplier. Using (6.10), Lemma 4.4 and h12 ≥ h1 , h2 ≥ h1 , k −
integral can be estimated from above by
Z∞
1
1
2
≥ 14 , this multiple
1
Φ(h1 )n−2d+k φ(h1 )d−k+ 2 φ(h1 )k− 2 φ(h1 )
1
Z∞ Z∞
×
d−k
φ(h2 )
S(h1 , h2 , ϕ)φ(−h12 )φ
h1 − h2 cos ϕ
2 sin ϕ
(sin ϕ)d−k−2 dϕ dh2 dh1
h1 0
Z∞
≤ c22
−(d−k+3)
Φ(h1 )n−2d+k φ(h1 )d+1 h1
φ(h1 )d−k dh1
1
≤ c23 n−2d+k−1 (ln n)
d−3
2
.
(7.6)
Thus, combining (7.3), (7.4), (7.5) and (7.6), we finally obtain
E (Vd (Pn+1 − Vd (Pn ))2 ≤ c24 n−1 (ln n)
d−3
2
,
which completes the proof.
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Authors’ addresses:
Daniel Hug, Mathematisches Institut, Albert-Ludwigs-Universität, Eckerstr. 1, D-79104 Freiburg i. Br.,
Germany
e-mail: [email protected]
Matthias Reitzner, Institut für Analysis und Technische Mathematik, Technische Universität Wien, Wiedner Hauptstrasse 8–10, A-1040 Vienna, Austria
e-mail: [email protected]
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