Mathematics authors titles recent submissions
RANDOM AFFINE SIMPLEXES
arXiv:1711.06578v1 [math.PR] 17 Nov 2017
FRIEDRICH GÖTZE, ANNA GUSAKOVA, AND DMITRY ZAPOROZHETS
Abstract. For a fixed k ∈ {1, . . . , d} consider random vectors X0 , . . . , Xk ∈ Rd with an
arbitrary spherically symmetric joint density function. Let A be any non-singular d × d
matrix. We show that the k-dimensional volume of the convex hull of affinely transformed
Xi ’s satisfies
d |Pξ E|
· | conv(X0 , . . . , Xk )|,
| conv(AX0 , . . . , AXk )| =
κk
where E := {x ∈ Rd : x⊤ (A⊤ A)−1 x ≤ 1} is an ellipsoid, Pξ denotes the orthogonal
projection to a random uniformly chosen k-dimensional linear subspace ξ independent of
X0 , . . . , Xk , and κk is the volume of the unit k-dimensional ball.
We express |Pξ E| in terms of Gaussian random matrices. The important special case
k = 1 corresponds to the distance between two random points:
s
λ21 N12 + · · · + λ2d Nd2
d
|AX0 − AX1 | =
· |X0 − X1 |,
N12 + · · · + Nd2
where N1 , . . . , Nd are i.i.d. standard Gaussian variables independent of X0 , X1 and λ1 , . . . , λd
are the singular values of A.
As an application, we derive the following integral geometry formula for ellipsoids:
Z
Z
κk(d+p)+k
κk+1
p+d+1
k+1
d
|E ∩ E|
µd,k (dE) = |E|
|PL E|p νd,k (dL),
κk(d+p)+d
κd+1
k
Ad,k
Gd,k
where p > −d + k − 1 and Ad,k and Gd,k are the affine and the linear Grassmannians
equipped with their respective Haar measures. The case p = 0 reduces to an affine version
of the integral formula of Furstenberg and Tzkoni.
1. Main results
1.1. Basic notation. First we introduce some basic notion of integral geometry following [18]. The Euclidean space Rd is equipped with the Euclidean scalar product h·, ·i. The
volume is denoted by | · |. Some of the sets we consider have dimension less than d. In
fact, we consider 3 classes: the convex hulls of k + 1 points, orthogonal projections to kdimensional linear subspaces, and intersections with k-dimensional affine subspaces, where
k ∈ {0, . . . , d}. In this case, | · | stands for k-dimensional volume.
2010 Mathematics Subject Classification. Primary, 60D05, 52A22; secondary, 60B20, 78A40, 52A39.
Key words and phrases. Blaschke-Petkantschin formula, convex hulls, ellipsoids, expected volume,
Furstenberg-Tzkoni formula, Gaussian matrices, intrinsic volumes, random sections, random simplexes.
The work was done with the financial support of the Bielefeld University (Germany) in terms of projects
SFB 1283 and IRTG 2235. The work of the third author is supported by the grant RFBR 16-01-00367 and by
the Program of Fundamental Researches of Russian Academy of Sciences “Modern Problems of Fundamental
Mathematics”.
1
2
FRIEDRICH GÖTZE, ANNA GUSAKOVA, AND DMITRY ZAPOROZHETS
The unit ball in Rk is denoted by Bk . For p > 0 we write
κp :=
π p/2
,
Γ 2p + 1
(1)
where for an integer k we have κk = |Bk |.
For k ∈ {0, . . . , d}, the linear (resp., affine) Grassmannian of k-dimensional linear (resp.,
affine) subspaces of Rd is denoted by Gd,k (resp., Ad,k ) and is equipped with a unique rotation
invariant (resp., rigid motion invariant) Haar measure νd,k (resp., µd,k ), normalized by
νd,k (Gd,k ) = 1,
respectively,
µd,k
E ∈ Ad,k : E ∩ Bd 6= ∅ = κd−k .
A compact convex subset K of Rd with non-empty interior is called a convex body. We
define the intrinsic volumes of K by Kubota’s formula:
Z
κd
d
Vk (K) =
|PL K| νd,k (dL),
(2)
k κk κd−k
Gd,k
where PL K denotes the image of K under the orthogonal projection to L.
For L ∈ Gd,k (resp., E ∈ Ad,k ) we denote by λL (resp., λE ) the k-dimensional Lebesgue
measures on L (resp., E ).
1.2. Affine transformation of spherically symmetric distribution. For a fixed k ∈
{1, . . . , d} consider random vectors X0 , . . . , Xk ∈ Rd (not necessary i.i.d.) with an arbitrary
spherically symmetric joint density function f . Almost surely, their convex hull
conv(X0 , . . . , Xk )
is a k-dimensional simplex with well-defined k-dimensional volume
| conv(X0 , . . . , Xk )|.
(3)
How does the volume (3) changes under affine transformations? For k = d, the answer is
obvious: it is multiplied by the determinant of the transformation. The case k < d presents
a more delicate problem.
Theorem 1.1. Let A be any non-singular d × d matrix. Denote by E the ellipsoid defined
by
E := x ∈ Rd : x⊤ (A⊤ A)−1 x ≤ 1 .
(4)
Then we have
d |Pξ E|
| conv(AX0 , . . . , AXk )| =
· | conv(X0 , . . . , Xk )|,
(5)
κk
where Pξ denotes the orthogonal projection to a random uniformly chosen k-dimensional
linear subspace ξ independent of X0 , . . . , Xk .
Due to Kubota’s formula (see (2)), E |Pξ E| is proportional to Vk (E). Thus, taking expectation in (5) readily implies the following corollary.
RANDOM AFFINE SIMPLEXES
Corollary 1.2. Under the assumptions of Theorem 1.1 we have
−1
κk κd−k
d
Vk (E) E | conv(X0 , . . . , Xk )|.
E | conv(AX0 , . . . , AXk )| =
κd
k
3
(6)
For a formula for Vk (E), see [11]. Relation (6) can be generalized to higher moments
using the notion of generalized intrinsic volumes introduced in [4], but we shall skip to
describe details here.
The main ingredients of the proof of Theorem 1.1 are the Blaschke-Petkantschin formula
(see Subsection 2.1) and the following deterministic version of (5).
Proposition 1.3. Consider x1 , . . . , xk ∈ Rd and denote by L the span (linear hull) of
0, x1 , . . . , xk . Then
|PL E|
| conv(0, Ax1 , . . . , Axk )| =
· | conv(0, x1 , . . . , xk )|.
(7)
κk
Let us stress that here the origin is added to the convex hull.
Applying (7) to standard Gaussian vectors (details are in Subsection 2.4) leads to the
following representation.
Corollary 1.4. Under the assumptions of Theorem 1.1 we have
!1/2
!1/2
⊤
⊤ ⊤
det
G
G
det
G
A
AG
|Pξ E| d
d
λ λ
=
=
,
⊤
κk
det G G
det G⊤ G
(8)
where G is a random d × k matrix with i.i.d. standard Gaussian entries Nij and Gλ is a
random d × k matrix with the entries λi Nij , where λ1 , . . . , λd denote the singular values of
A.
Thus, we obtain the following version of (5).
Corollary 1.5. Under the assumptions of Theorem 1.1 we have
!1/2
det G⊤ A⊤ AG
d
| conv(AX0 , . . . , AXk )| =
| conv(X0 , . . . , Xk )|
det G⊤ G
!1/2
⊤
det
G
G
d
λ λ
| conv(X0 , . . . , Xk )|.
=
det G⊤ G
The important special case k = 1 corresponds to the distance between two random
points.
Corollary 1.6. Under the assumptions of Theorem 1.1 we have
s
λ21 N12 + · · · + λ2d Nd2
d
|AX0 − AX1 | =
· |X0 − X1 |,
N12 + · · · + Nd2
where N1 , . . . , Nd are i.i.d. standard Gaussian variables.
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FRIEDRICH GÖTZE, ANNA GUSAKOVA, AND DMITRY ZAPOROZHETS
These results will be used in the next subsection to study integral geometry problems
for ellipsoids.
1.3. Random points in ellipsoids. Now suppose that X0 , . . . , Xk are uniformly distributed
in some convex set K ⊂ Rd with non-empty interior. A classical problem of stochastic geometry is to find the distribution of (3) starting with its moments
Z
1
p
E | conv(X0 , . . . , Xk )| =
| conv(x0 , . . . , xk )|p dx0 . . . dxk .
(9)
|K|k+1
K k+1
To the best of our knowledge, for general K a formula for (9) is not known even for
d = 2, k = p = 1, where the problem reduces to the calculating the mean distance between
two random points uniformly chosen in a planar convex set (see [2], [6], [17, Chapter 4], [14,
Chapter 2], [1]).
For an arbitrary d and k = 1, there is an electromagnetic interpretation of (9) (see [8]): a
transmitter X0 and a receiver X1 are placed random uniformly in K. It is empirically known
that the power received decreases with an inverse distance law of the form 1/|X0 − X1 |α ,
where α is the so-called path-loss exponent, which depends on the environment in which
both are located (see [16]). Thus (9) with k = 2 and p = −nα expresses the n-th moment
of the power received (n < d/α).
The case of arbitrary k and d was studied when K is a ball only. In [15] it was shown
(see also [18, Theorem 8.2.3]) that for X0 , . . . , Xk uniformly distributed in the unit ball Bd
in Rd and integer p ≥ 0
κk+1
bd,k
d+p κk(d+p)+d
E | conv(X0 , . . . , Xk )| = k+1
,
κd κ(k+1)(d+p) bd+p,k
p
where κk are are defined in (1) and for any real number q > k − 1 we write
κq−k+1 · · · κq
bq,k := (k!)q−k−1 (q − k + 1) · · · q ·
.
κ1 · · · κk
(10)
(11)
In [12] this relation was extended to all real p > −d + k − 1. Theorem 1.1 implies (for details
see Subsection 3.1) the following generalization of (10) for ellipsoids. Recall that Pξ denotes
the orthogonal projection to a random uniformly chosen k-dimensional linear subspace ξ
independent with X0 , . . . , Xk
Theorem 1.7. For X0 , . . . , Xk uniformly distributed in any non-degenerate ellipsoid E ⊂ Rd
and any real number p > −d + k − 1 we have
κk+1
bd,k E |Pξ E|p
d+p κk(d+p)+d
E | conv(X0 , . . . , Xk )| = k+1
.
κpk
κd κ(k+1)(d+p) bd+p,k
p
(12)
Note that (12) is indeed a generalization of (10) since Pξ Bd = Bk a.s. and |Bk |p = κpk .
For k = 1 formula (12) was recently obtained in [9].
For p = 1, the right-hand side of (12) is proportional to the k-th intrinsic volume of E
(see (2)), which implies the following result (for details see Subsection 3.2).
RANDOM AFFINE SIMPLEXES
5
Corollary 1.8. For X0 , . . . , Xk uniformly distributed in any non-degenerate ellipsoid E ⊂ Rd
we have
!2
κk+1
1 ((d + 1)!)k+1
d+1
Vk (E).
E | conv(X0 , . . . , Xk )| = k
2 ((d + 1)(k + 1))! κ(d+1)(k+1)
Very recently, for X0 , . . . , Xk uniformly distributed in the unit ball Bd , a formula for
the distribution of | conv(X0 , . . . , Xk )| has been derived in [7]. For a random variable η and
α1 , α2 > 0 we write η ∼ B(α1 , α2 ) to denote that η has a Beta distribution with parameters
α1 , α2 and the density
Γ(α1 + α2 ) α1 −1
t
(1 − t)α2 −1 ,
Γ(α1 ) Γ(α2 )
t ∈ (0, 1).
It was shown in [7] that for X0 , . . . , Xk uniformly distributed in Bd ,
d
(k!)2 η(1 − η)k | conv(X0 , . . . , Xk )|2 = (1 − η ′ )k η1 · · · ηk ,
(13)
where η, η ′ , η1 , . . . , ηk are independent random variables independent with X0 , . . . , Xk such
that
d−k+i k−i
kd
d
′
, ηi ∼ B
+ 1,
,
+1 .
η, η ∼ B
2
2
2
2
Multiplying both sides of (13) by |Pξ E|2 /κ2k and applying Theorem 1.1 and Corollary 1.4
(for details see Subsection 3.1) leads to the following generalization of (13).
Theorem 1.9. For X0 , . . . , Xk uniformly distributed in any non-degenerate ellipsoid E ⊂ Rd
we have
d
′ k
2
(k!)2 η (1 − η)k | conv(X0 , . . . , Xk )|2 = κ−2
k (1 − η ) η1 · · · ηk |Pξ E|
d
= (1 − η ′ )k η1 · · · ηk
!
det G⊤
λ Gλ
,
det (G⊤ G)
where the matrices G and Gλ are defined in Corollary 1.4 and λ1 , . . . , λd denoting the length
of semi-axes of E.
Taking k = 1 yields the distribution of the distance between two random points in E.
Corollary 1.10. Under the assumptions of Theorem 1.9 we have
2 2
λ1 N1 + · · · + λ2d Nd2
2 d
′
η(1 − η) · |X0 − X1 | = (1 − η ) η1
,
N12 + · · · + Nd2
where N1 , . . . , Nd are i.i.d. standard Gaussian variables.
6
FRIEDRICH GÖTZE, ANNA GUSAKOVA, AND DMITRY ZAPOROZHETS
1.4. Integral geometry formulas. Recall that Gd,k and Ad,k denote the linear and affine
Grassmannians defined in Subsection 1.1.
For an arbitrary convex compact body K, p ∈ Z+ , and k = 1 it is possible to express (9)
in terms of the lengths of the one-dimensional sections of K (see [3] and [13]):
Z
Z
2dκd
p
|K ∩ E|p+d+1 µd,1 (dE).
|x0 − x1 | dx0 dx1 =
(d + p) (d + p + 1)
Ad,1
K2
This formula does not extend to k > 1. For ellipsoids K = E this is possible (for details see
Subsection 4.1).
Theorem 1.11. For any non-degenerate ellipsoid E ⊂ Rd , k ∈ {0, 1, . . . , d}, and any real
number p > −d + k − 1 we have
Z
Z
κk+1
κk(d+p)+k bd,k
d+p
p
|E ∩ E|p+d+1 µd,k (dE).
| conv(x0 , . . . , xk )| dx0 . . . dxk = p+d+1
κ
b
κk
(k+1)(d+p) d+p,k
Ad,k
E k+1
(14)
Comparing this theorem with Theorem 1.7 readily gives the following connection between the volumes of k-dimensional cross-sections and projections of ellipsoids.
Theorem 1.12. Under the assumptions of Theorem 1.11 we have
Z
Z
κk(d+p)+k
κk+1
p+d+1
k+1
d
|E ∩ E|
µd,k (dE) = |E|
|PL E|p νd,k (dL).
d+1 κ
κk
k(d+p)+d
Ad,k
Gd,k
For p = 0, we obtain the following integral formula.
Corollary 1.13. Under the assumptions of Theorem 1.11 we have
Z
κd(k+1)
κd+1
k
|E ∩ E|d+1 µd,k (dE) = k+1
|E|k+1.
κd κk(d+1)
(15)
Ad,k
This result may be regarded as an affine version of the following integral formula of
Furstenberg and Tzkoni (see [5]):
Z
κd
|E ∩ L|d νd,k (dL) = kk |E|k .
κd
Gd,k
Our next theorem generalizes this formula in the same way as Theorem 1.11 generalizes (15).
Theorem 1.14. For any non-degenerate ellipsoid E ⊂ Rd , k ∈ {0, 1, . . . , d}, and any real
number p > −d + k we have
Z
Z
κkd+p bd,k
p
| conv(0, x1 , . . . , xk )| dx1 . . . dxk = p+d
|E ∩ L|p+d νd,k (dL).
(16)
κk bd+p,k
Ek
Gd,k
RANDOM AFFINE SIMPLEXES
7
In probabilistic language it may be formulated as
κkd+p bd,k
E | conv(0, X1 , . . . , Xk )| = p+d
E |E ∩ ξ|p+d,
κk bd+p,k
p
where X1 , . . . , Xk are i.i.d. random vectors uniformly distributed in E and ξ is a uniformly
chosen random k-dimensional linear subspace in Rd .
2. Proofs: Part I
2.1. Blaschke-Petkantschin formula. In our calculations we will need to integrate some
non-negative measurable function h of k-tuples of points in Rd . To this end, we integrate
first over the k-tuples of points in a fixed k-dimensional linear subspace L, with respect to
the product measure λkL , and then integrate over Gd,k , with respect to νd,k . The corresponding transformation formula is known as the linear Blaschke-Petkantschin formula (see [18,
Theorem 7.2.1]):
Z
h(x1 , . . . , xk ) dx1 . . . dxk =
(17)
(Rd )k
bd,k
Z Z
h(x1 , . . . , xk ) | conv(0, x1 , . . . , xk )|d−k λL (dx1 ) . . . λL (dxk ) νd,k (dL),
Gd,k Lk
where bd,k is defined in (11).
A similar affine version (see [18, Theorem 7.2.7]) may be stated as follows:
Z
h(x0 , . . . , xk ) dx0 . . . dxk =
(18)
(Rd )k+1
bd,k
Z
Z
h(x0 , . . . , xk ) | conv(x0 , . . . , xk )|d−k λE (dx0 ) . . . λE (dxk ) µd,k (dE).
Ad,k E k+1
2.2. Proof of Proposition 1.3. To avoid trivialities we assume that dim L = k. Let
e1 , . . . , ek ∈ Rd be some orthonormal basis in L. Let OL and X denote d × k matrices whose
columns are e1 , . . . , ek and x1 , . . . , xk respectively. It is easy to check that OL OL⊤ is a d × d
matrix corresponding to the orthogonal projection operator PL . Thus,
OL OL⊤ X = X.
(19)
Recall that E is defined by (4). It is known (see, e.g., [19, Appendix H]) that the
orthogonal projection PL E is an ellipsoid in L and
h
i1/2
|PL E| = κk det OL⊤ HOL
,
(20)
where
H := A⊤ A.
8
FRIEDRICH GÖTZE, ANNA GUSAKOVA, AND DMITRY ZAPOROZHETS
A well-known formula for the volume of a k-dimensional simplex states that for any x1 , . . . , xk ∈
Rd ,
i1/2
1h
det X ⊤ X
.
(21)
| conv(0, x1 , . . . , xk )| =
k!
Therefore,
h
i1/2 h
i1/2
k! | conv(0, Ax1 , . . . , Axk )| = det (AX)⊤ AX
= det X ⊤ HX
.
Applying (19) produces
det X ⊤ HX = det X ⊤ OL OL⊤ HOL OL⊤ X = det OL⊤ HOL det X ⊤ OL det OL⊤ X
= det OL⊤ HOL det X ⊤ OL OL⊤ X = det OL⊤ HOL det X ⊤ X ,
which together with (20) and (21) finishes the proof.
2.3. Proof of Theorem 1.1. Denote by f (x0 , . . . , xk ) the joint density of (X0 , . . . , Xk ).
Let
Z
ϕA (t) :=
exp it log | conv(Ax0 , . . . , Axk )| f (x0 , . . . , xk ) dx0 . . . dxk
(Rd )k+1
=
Z
(Rd )k+1
exp it log | conv(0, A(x1 − x0 ), . . . , A(xk − x0 ))| f (x0 , . . . , xk ) dx0 . . . dxk
be a characteristic function of log | conv(AX0 , . . . , AXk )|. In particular, denoting by I the
identity matrix, we obtain that ϕI (t) is a characteristic function of log | conv(X0 , . . . , Xk )|.
Substituting y0 = x0 and yi = xi − x0 for 1 ≤ i ≤ k leads to
Z
ϕA (t) =
exp it log | conv(0, Ay1 , . . . , Ayk )| f (y0 , y1 + y0 . . . , yk + y0 ) dy0 . . . dyk
(Rd )k+1
=
Z
(Rd )k
exp it log | conv(0, Ay1 , . . . , Ayk )| g(y1 , . . . , yk ) dy1 . . . dyk ,
where
g(y1 , . . . , yk ) :=
Z
f (y0 , y1 + y0 . . . , yk + y0 ) dy0 .
Rd
Using the linear Blaschke-Petkantschin formula (see (17)) with
gives
ϕA (t) =
h(y1 , . . . , yk ) := exp it log | conv(0, Ay1 , . . . , Ayk )| g(y1 , . . . , yk )
Z Z
Gd,k Lk
exp it log | conv(0, Ay1 , . . . , Ayk )| g(y1 , . . . , yk )×
| conv(0, y1 , . . . , yk )|d−k λL (dy1 ) . . . λL (dyk ) νd,k (dL). (22)
RANDOM AFFINE SIMPLEXES
9
Applying Proposition 1.3 to (22) gives
Z
Z
|PL E|
exp it log | conv(0, y1 , . . . , yk )| g(y1 , . . . , yk )
ϕA (t) =
exp it log
κk
Gd,k
Lk
× | conv(0, y1 , . . . , yk )|d−k λL (dy1 ) . . . λL (dyk ) νd,k (dL).
Since f is spherically invariant, the function
Z
hA (t) := exp it log | conv(0, y1 , . . . , yk )| g(y1 , . . . , yk )
Lk
× | conv(0, y1 , . . . , yk )|d−k λL (dy1 ) . . . λL (dyk )
does not depend on the choice of L. Indeed, consider any L′ ∈ Gd,k . There exists an
orthogonal matrix U such that L = UL′ . Substituting yi = Uzi gives
Z
hA (t) =
exp it log | conv(0, Uz1 , . . . , Uzk )| g(Uz1 , . . . , Uzk )
L′ k
× | conv(0, Uz1 , . . . , Uzk )|d−k λ′L (dz1 ) . . . λ′L (dzk ).
Now the claim follows from
| conv(0, Uz1 , . . . , Uzk )| = | conv(0, z1 , . . . , zk )|
and
g(Uz1 , . . . , Uzk ) =
Z
f (y0 , Uz1 + y0 . . . , Uzk + y0 ) dy0
Z
f (Uy0 , Uz1 + Uy0 . . . , Uzk + Uy0 ) dy0
Z
f (y0 , z1 + y0 . . . , zk + y0 ) dy0
Rd
=
Rd
=
Rd
= g(z1 , . . . , zk ),
where at the second step we did a variable change y0 → Uy0 and at the third step we used
the spherical symmetry of f .
Thus hA (t) does not depend on the choice of L, which implies
|Pξ E|
ϕA (t) = hA (t) E exp it log
.
κk
In particular,
ϕI (t) = hA (t).
10
FRIEDRICH GÖTZE, ANNA GUSAKOVA, AND DMITRY ZAPOROZHETS
Comparing the last two equalities and applying the characteristic function uniqueness theorem, we arrive at
d
log | conv(AX0 , . . . , AXk )| = log
|Pξ E|
+ log | conv(X0 , . . . , Xk )|,
κk
and the theorem follows.
2.4. Proof of Corollary 1.4. Denote by G1 , . . . , Gk ∈ Rd the columns of the matrix G.
Hence, AG1 , . . . , AGk ∈ Rd are the columns of the matrix AG. Using Proposition 1.3 with
xi = Gi and applying (21) to G and AG gives
h
or
⊤
⊤
det G A AG
i1/2
i1/2
|Pη E| h
⊤
· det G G
,
=
κk
det G⊤ A⊤ AG
det G⊤ G
!1/2
=
|Pη E|
,
κk
where η is the linear hull of 0, G1 , . . . , Gk . Since G1 , . . . , Gk are i.i.d. standard Gaussian
d
vectors, η is uniformly distributed in Gd,k with respect to νd,k , which implies η = ξ, and the
corollary follows.
3. Proofs: Part II
3.1. Proofs of Theorem 1.7 and Theorem 1.9. For any non-degenerate ellipsoid E there
exist a unique symmetric positive-definite d × d matrix A such that
E = ABd = x ∈ Rd : kA−1 xk ≤ 1 = x ∈ Rd : x⊤ A−2 x ≤ 1 .
Since X0 , . . . , Xk are i.i.d. random vectors uniformly distributed in E, we have that A−1 X0 ,
. . . , A−1 Xk are i.i.d. random vectors uniformly distributed in Bd . It follows from Theorem 1.1
that
d
d
| conv(X0 , . . . , Xk )| = | conv(AA−1 X0 , . . . , AA−1 Xk )| = | conv(A−1 X0 , . . . , A−1 Xk )|
Taking the p-th moment and applying (10) implies Theorem 1.7.
Now apply (13) to A−1 X0 , . . . , A−1 Xk :
d
(k!)2 η(1 − η)k | conv(A−1 X0 , . . . , A−1 Xk )|2 = (1 − η ′ )k η1 · · · ηk .
Multiplying by
|Pξ E|
κpk
and applying (23) implies Theorem 1.9.
|Pξ E|
.
κpk
(23)
RANDOM AFFINE SIMPLEXES
11
3.2. Proof of Corollary 1.8. From Kubota’s formula (see (2)) and Theorem 1.7 we have
E | conv(X0 , . . . , Xk )| = αd,k Vk (E),
where
κk+1
bd,k κd−k
d+1 κk(d+1)+d
.
d
k+1 κ
κd
(k+1)(d+1) bd+1,k k κd
(see (11)) and κp (see (1)) we obtain
αd,k :=
From the definition of bd,k
κk+1
d+1 κk(d+1)+d (d + 1 − k)! κd−k+1 κd−k
κ(k+1)(d+1)
(d + 1)! κd+1
κd
κk+1
d
!k+1
Γ 12 d + 1
(d + 1 − k)!
= k/2
π (d + 1)! Γ 21 (d + 1) + 1
Γ 21 (d + 1) + 1
Γ 21 d + 1
Γ 21 (k + 1)(d + 1) + 1
.
×
Γ 21 ((k + 1)d + k) + 1 Γ 21 (d − k + 1) + 1 Γ 12 (d − k) + 1
αd,k =
Using Legendre’s duplication formula for the Gamma function
1
= 21−2z π 1/2 Γ (2z) ,
Γ (z) Γ z +
2
the recursion Γ (1 + z) = z Γ (z), and the fact that k, d ∈ Z we obtain
!k+1
Γ 21 d + 1
Γ 12 d + 21 Γ 12 d + 1
(d − k)! Γ 21 (k + 1)(d + 1) + 1
αd,k = k/2
π d! Γ 12 ((k + 1)d + k) + 1 Γ 21 (d − k) + 21 Γ 12 (d − k) + 1
Γ 12 (d + 1) + 1
!k+1
Γ 12 (k + 1)(d + 1) + 1
Γ 21 d + 1
1
= √ k
1
Γ 12 (d + 1) + 1
(2 π) Γ 2 ((k + 1)d + k) + 1
!2
k+1
κk+1
Γ 12 d + 1 Γ 12 d + 1 + 12
1
d+1
= √ k
1
1
1
κ
Γ
(kd
+
d
+
k)
+
1
Γ
(kd
+
k
+
d)
+
1
+
(d+1)(k+1)
(2 π)
2
2
2
!2
κk+1
1 ((d + 1)!)k+1
d+1
= k
.
2 ((d + 1)(k + 1))! κ(d+1)(k+1)
4. Proofs: Part III
4.1. Proof of Theorem 1.11. Let
Z
J :=
| conv(x0 , . . . , xk )|p dx0 . . . dxk =
E k+1
Z
| conv(x0 , . . . , xk )|p
(Rd )k+1
Using the affine Blaschke-Petkantschin formula (see (18)) with
h(x0 , . . . , xk ) := | conv(x0 , . . . , xk )|p
k
Y
i=0
1E (xi )
k
Y
i=0
1E (xi ) dx0 . . . dxk .
12
FRIEDRICH GÖTZE, ANNA GUSAKOVA, AND DMITRY ZAPOROZHETS
yields
J = bd,k
Z
Z
| conv(x0 , . . . , xk )|
p+d−k
= bd,k
Z
1E (xi ) λE (dx0 ) . . . λE (dxk ) µd,k (dE)
i=0
Ad,k E k+1
Z
k
Y
| conv(x0 , . . . , xk )|p+d−k λE (dx0 ) . . . λE (dxk ) µd,k (dE).
Ad,k (E∩E)k+1
Now fix E ∈ Ad,k . Applying Theorem 1.7 to the ellipsoid E ∩ E gives
1
|E ∩ E|k+1
Z
| conv(x0 , . . . , xk )|p+d−k λE (dx0 ) . . . λE (dxk )
(E∩E)k+1
=
κk+1
d+p
κp+d+1
k
κk(d+p)+k
1
|E ∩ E|p+d−k ,
κ(k+1)(d+p) bd+p,k
which leads to
J=
κk+1
d+p
κp+d+1
k
κk(d+p)+k bd,k
κ(k+1)(d+p) bd+p,k
Z
|E ∩ E|p+d+1 µd,k (dE).
Ad,k
4.2. Proof of Theorem 1.14. The proof is similar to the previous one. Let
J :=
Z
p
| conv(0, x1 , . . . , xk )| dx1 . . . dxk =
Ek
Z
| conv(0, x1 , . . . , xk )|
(Rd )k
p
k
Y
1E (xi ) dx1 . . . dxk .
i=1
Using the linear Blaschke-Petkantschin formula (see (17)) with
h(x1 , . . . , xk ) := | conv(0, x1 , . . . , xk )|
p
k
Y
1E (xi )
i=1
gives
J = bd,k
Z Z
| conv(0, x1 , . . . , xk )|
Gd,k Lk
= bd,k
Z
Z
Gd,k (L∩E)k
p+d−k
k
Y
1E (xi ) λL (dx1 ) . . . λL (dxk ) νd,k (dL)
i=1
| conv(0, x1 , . . . , xk )|p+d−k λL (dx1 ) . . . λL (dxk ) νd,k (dL). (24)
RANDOM AFFINE SIMPLEXES
13
Fix L ∈ Gd,k . Since E ∩ L is an ellipsoid, there exists a linear transform AL : L → Rk such
that AL (E ∩ L) = Bk . Applying the coordinate transform xi = AL yi , i = 1, 2, . . . , k, we get
Z
| conv(0, x1 , . . . , xk )|p+d−k λL (dx1 ) . . . λL (dxk )
(L∩E)k
=
|E ∩ L|p+d
κp+d
k
Z
| conv(0, y1 , . . . , yk )|p+d−k λL (dy1 ) . . . λL (dyk ). (25)
(Bk )k
It is known (see, e.g., [18, Theorem 8.2.2]) that
Z
k
Y
p+d−k
−p−d+k k
| conv(0, y1 , . . . , yk )|
λL (dy1 ) . . . λL (dyk ) = (k!)
κd+p
i=1
(Bk )k
(k + 1 − i) κk+1−i
.
(d + p + 1 − i) κd+p+1−i
(26)
Substituting (26) and (25) into (24) finishes the proof.
4.3. Acknowledgements. The authors are grateful to Günter Last and Daniel Hug for
helpful discussions and suggestions which improved this paper.
References
[1] U. Bäsel. Random chords and point distances in regular polygons. Acta Math. Univ.
Comenianae, 83(1):1–18, 2014.
[2] É. Borel. Principes et formules classiques du calcul des probabilités. Gauthier-Villars,
Paris, 1925.
[3] G. Chakerian. Inequalities for the difference body of a convex body. Proc. Amer. Math.
Soc., 18(5):879–884, 1967.
[4] N. Dafnis and G. Paouris. Estimates for the affine and dual affine quermassintegrals of
convex bodies. Illinois J. Math., 56(4):1005–1021, 2012.
[5] H. Furstenberg and I. Tzkoni. Spherical functions and integral geometry. Israel J. Math.,
10(3):327–338, 1971.
[6] B. Ghosh. Random distances within a rectangle and between two rectangles. Bull.
Calcutta Math. Soc., 43:17–24, 1951.
[7] J. Grote, Z. Kabluchko, and C. Täle. Limit theorems for random simplices in high
dimensions. Preprint, arXiv:1708.00471, 2017.
[8] J. Hansen and M. Reitzner. Electromagnetic wave propagation and inequalities for
moments of chord lengths. Adv. in Appl. Probab., 36(04):987–995, 2004.
[9] L. Heinrich. Lower and upper bounds for chord power integrals of ellipsoids. App. Math.
Sci., 8(165):8257–8269, 2014.
[10] R. Horn and C. Johnson. Matrix analysis. Cambridge university press, 2012.
[11] Z. Kabluchko and D. Zaporozhets. Random determinants, mixed volumes of ellipsoids,
and zeros of Gaussian random fields. J. Math. Sci., 199(2):168–173, 2014.
[12] Z. Kabluchko, D. Temesvari, and C. Täle. Expected intrinsic volumes and facet numbers
of random beta-polytopes. Preprint, arXiv:1707.02253, 2017.
14
FRIEDRICH GÖTZE, ANNA GUSAKOVA, AND DMITRY ZAPOROZHETS
[13] J. Kingman. Random secants of a convex body. J. Appl. Probab., 6(3):660–672, 196.
[14] A. Mathai. An introduction to geometrical probability: distributional aspects with applications, volume 1 of Statistical Distributions and Models with Applications. Gordon and
Breach Science Publishers, Amsterdam, 1999.
[15] R. Miles. Isotropic random simplices. Adv. in Appl. Probab., 3(02):353–382, 1971.
[16] T. Rappaport, A. Annamalai, R. Buehrer, and W. Tranter. Wireless communications:
past events and a future perspective. IEEE Commun. Mag., 40(5):148–161, 2002.
[17] L. Santaló. Integral Geometry and Geometric Probability. Addison-Wesley Publishing
Company, 1976.
[18] R. Schneider and W. Weil. Stochastic and integral geometry. Springer–Verlag, Berlin,
2008.
[19] F. C. Schweppe. Uncertain Dynamic Systems. Prentice-Hall, Englewood Cliffs, NJ,
1973.
Friedrich Götze: Faculty of Mathematics, Bielefeld University, P. O. Box 10 01 31,
33501 Bielefeld, Germany
E-mail address: goetze@math.uni-bielefeld.de
Anna Gusakova: Faculty of Mathematics, Bielefeld University, P. O. Box 10 01 31,
33501 Bielefeld, Germany
E-mail address: agusakov@math.uni-bielefeld.de
Dmitry Zaporozhets: St. Petersburg Department of Steklov Mathematical Institute,
Fontanka 27, 191023 St. Petersburg, Russia
E-mail address: zap1979@gmail.com
arXiv:1711.06578v1 [math.PR] 17 Nov 2017
FRIEDRICH GÖTZE, ANNA GUSAKOVA, AND DMITRY ZAPOROZHETS
Abstract. For a fixed k ∈ {1, . . . , d} consider random vectors X0 , . . . , Xk ∈ Rd with an
arbitrary spherically symmetric joint density function. Let A be any non-singular d × d
matrix. We show that the k-dimensional volume of the convex hull of affinely transformed
Xi ’s satisfies
d |Pξ E|
· | conv(X0 , . . . , Xk )|,
| conv(AX0 , . . . , AXk )| =
κk
where E := {x ∈ Rd : x⊤ (A⊤ A)−1 x ≤ 1} is an ellipsoid, Pξ denotes the orthogonal
projection to a random uniformly chosen k-dimensional linear subspace ξ independent of
X0 , . . . , Xk , and κk is the volume of the unit k-dimensional ball.
We express |Pξ E| in terms of Gaussian random matrices. The important special case
k = 1 corresponds to the distance between two random points:
s
λ21 N12 + · · · + λ2d Nd2
d
|AX0 − AX1 | =
· |X0 − X1 |,
N12 + · · · + Nd2
where N1 , . . . , Nd are i.i.d. standard Gaussian variables independent of X0 , X1 and λ1 , . . . , λd
are the singular values of A.
As an application, we derive the following integral geometry formula for ellipsoids:
Z
Z
κk(d+p)+k
κk+1
p+d+1
k+1
d
|E ∩ E|
µd,k (dE) = |E|
|PL E|p νd,k (dL),
κk(d+p)+d
κd+1
k
Ad,k
Gd,k
where p > −d + k − 1 and Ad,k and Gd,k are the affine and the linear Grassmannians
equipped with their respective Haar measures. The case p = 0 reduces to an affine version
of the integral formula of Furstenberg and Tzkoni.
1. Main results
1.1. Basic notation. First we introduce some basic notion of integral geometry following [18]. The Euclidean space Rd is equipped with the Euclidean scalar product h·, ·i. The
volume is denoted by | · |. Some of the sets we consider have dimension less than d. In
fact, we consider 3 classes: the convex hulls of k + 1 points, orthogonal projections to kdimensional linear subspaces, and intersections with k-dimensional affine subspaces, where
k ∈ {0, . . . , d}. In this case, | · | stands for k-dimensional volume.
2010 Mathematics Subject Classification. Primary, 60D05, 52A22; secondary, 60B20, 78A40, 52A39.
Key words and phrases. Blaschke-Petkantschin formula, convex hulls, ellipsoids, expected volume,
Furstenberg-Tzkoni formula, Gaussian matrices, intrinsic volumes, random sections, random simplexes.
The work was done with the financial support of the Bielefeld University (Germany) in terms of projects
SFB 1283 and IRTG 2235. The work of the third author is supported by the grant RFBR 16-01-00367 and by
the Program of Fundamental Researches of Russian Academy of Sciences “Modern Problems of Fundamental
Mathematics”.
1
2
FRIEDRICH GÖTZE, ANNA GUSAKOVA, AND DMITRY ZAPOROZHETS
The unit ball in Rk is denoted by Bk . For p > 0 we write
κp :=
π p/2
,
Γ 2p + 1
(1)
where for an integer k we have κk = |Bk |.
For k ∈ {0, . . . , d}, the linear (resp., affine) Grassmannian of k-dimensional linear (resp.,
affine) subspaces of Rd is denoted by Gd,k (resp., Ad,k ) and is equipped with a unique rotation
invariant (resp., rigid motion invariant) Haar measure νd,k (resp., µd,k ), normalized by
νd,k (Gd,k ) = 1,
respectively,
µd,k
E ∈ Ad,k : E ∩ Bd 6= ∅ = κd−k .
A compact convex subset K of Rd with non-empty interior is called a convex body. We
define the intrinsic volumes of K by Kubota’s formula:
Z
κd
d
Vk (K) =
|PL K| νd,k (dL),
(2)
k κk κd−k
Gd,k
where PL K denotes the image of K under the orthogonal projection to L.
For L ∈ Gd,k (resp., E ∈ Ad,k ) we denote by λL (resp., λE ) the k-dimensional Lebesgue
measures on L (resp., E ).
1.2. Affine transformation of spherically symmetric distribution. For a fixed k ∈
{1, . . . , d} consider random vectors X0 , . . . , Xk ∈ Rd (not necessary i.i.d.) with an arbitrary
spherically symmetric joint density function f . Almost surely, their convex hull
conv(X0 , . . . , Xk )
is a k-dimensional simplex with well-defined k-dimensional volume
| conv(X0 , . . . , Xk )|.
(3)
How does the volume (3) changes under affine transformations? For k = d, the answer is
obvious: it is multiplied by the determinant of the transformation. The case k < d presents
a more delicate problem.
Theorem 1.1. Let A be any non-singular d × d matrix. Denote by E the ellipsoid defined
by
E := x ∈ Rd : x⊤ (A⊤ A)−1 x ≤ 1 .
(4)
Then we have
d |Pξ E|
| conv(AX0 , . . . , AXk )| =
· | conv(X0 , . . . , Xk )|,
(5)
κk
where Pξ denotes the orthogonal projection to a random uniformly chosen k-dimensional
linear subspace ξ independent of X0 , . . . , Xk .
Due to Kubota’s formula (see (2)), E |Pξ E| is proportional to Vk (E). Thus, taking expectation in (5) readily implies the following corollary.
RANDOM AFFINE SIMPLEXES
Corollary 1.2. Under the assumptions of Theorem 1.1 we have
−1
κk κd−k
d
Vk (E) E | conv(X0 , . . . , Xk )|.
E | conv(AX0 , . . . , AXk )| =
κd
k
3
(6)
For a formula for Vk (E), see [11]. Relation (6) can be generalized to higher moments
using the notion of generalized intrinsic volumes introduced in [4], but we shall skip to
describe details here.
The main ingredients of the proof of Theorem 1.1 are the Blaschke-Petkantschin formula
(see Subsection 2.1) and the following deterministic version of (5).
Proposition 1.3. Consider x1 , . . . , xk ∈ Rd and denote by L the span (linear hull) of
0, x1 , . . . , xk . Then
|PL E|
| conv(0, Ax1 , . . . , Axk )| =
· | conv(0, x1 , . . . , xk )|.
(7)
κk
Let us stress that here the origin is added to the convex hull.
Applying (7) to standard Gaussian vectors (details are in Subsection 2.4) leads to the
following representation.
Corollary 1.4. Under the assumptions of Theorem 1.1 we have
!1/2
!1/2
⊤
⊤ ⊤
det
G
G
det
G
A
AG
|Pξ E| d
d
λ λ
=
=
,
⊤
κk
det G G
det G⊤ G
(8)
where G is a random d × k matrix with i.i.d. standard Gaussian entries Nij and Gλ is a
random d × k matrix with the entries λi Nij , where λ1 , . . . , λd denote the singular values of
A.
Thus, we obtain the following version of (5).
Corollary 1.5. Under the assumptions of Theorem 1.1 we have
!1/2
det G⊤ A⊤ AG
d
| conv(AX0 , . . . , AXk )| =
| conv(X0 , . . . , Xk )|
det G⊤ G
!1/2
⊤
det
G
G
d
λ λ
| conv(X0 , . . . , Xk )|.
=
det G⊤ G
The important special case k = 1 corresponds to the distance between two random
points.
Corollary 1.6. Under the assumptions of Theorem 1.1 we have
s
λ21 N12 + · · · + λ2d Nd2
d
|AX0 − AX1 | =
· |X0 − X1 |,
N12 + · · · + Nd2
where N1 , . . . , Nd are i.i.d. standard Gaussian variables.
4
FRIEDRICH GÖTZE, ANNA GUSAKOVA, AND DMITRY ZAPOROZHETS
These results will be used in the next subsection to study integral geometry problems
for ellipsoids.
1.3. Random points in ellipsoids. Now suppose that X0 , . . . , Xk are uniformly distributed
in some convex set K ⊂ Rd with non-empty interior. A classical problem of stochastic geometry is to find the distribution of (3) starting with its moments
Z
1
p
E | conv(X0 , . . . , Xk )| =
| conv(x0 , . . . , xk )|p dx0 . . . dxk .
(9)
|K|k+1
K k+1
To the best of our knowledge, for general K a formula for (9) is not known even for
d = 2, k = p = 1, where the problem reduces to the calculating the mean distance between
two random points uniformly chosen in a planar convex set (see [2], [6], [17, Chapter 4], [14,
Chapter 2], [1]).
For an arbitrary d and k = 1, there is an electromagnetic interpretation of (9) (see [8]): a
transmitter X0 and a receiver X1 are placed random uniformly in K. It is empirically known
that the power received decreases with an inverse distance law of the form 1/|X0 − X1 |α ,
where α is the so-called path-loss exponent, which depends on the environment in which
both are located (see [16]). Thus (9) with k = 2 and p = −nα expresses the n-th moment
of the power received (n < d/α).
The case of arbitrary k and d was studied when K is a ball only. In [15] it was shown
(see also [18, Theorem 8.2.3]) that for X0 , . . . , Xk uniformly distributed in the unit ball Bd
in Rd and integer p ≥ 0
κk+1
bd,k
d+p κk(d+p)+d
E | conv(X0 , . . . , Xk )| = k+1
,
κd κ(k+1)(d+p) bd+p,k
p
where κk are are defined in (1) and for any real number q > k − 1 we write
κq−k+1 · · · κq
bq,k := (k!)q−k−1 (q − k + 1) · · · q ·
.
κ1 · · · κk
(10)
(11)
In [12] this relation was extended to all real p > −d + k − 1. Theorem 1.1 implies (for details
see Subsection 3.1) the following generalization of (10) for ellipsoids. Recall that Pξ denotes
the orthogonal projection to a random uniformly chosen k-dimensional linear subspace ξ
independent with X0 , . . . , Xk
Theorem 1.7. For X0 , . . . , Xk uniformly distributed in any non-degenerate ellipsoid E ⊂ Rd
and any real number p > −d + k − 1 we have
κk+1
bd,k E |Pξ E|p
d+p κk(d+p)+d
E | conv(X0 , . . . , Xk )| = k+1
.
κpk
κd κ(k+1)(d+p) bd+p,k
p
(12)
Note that (12) is indeed a generalization of (10) since Pξ Bd = Bk a.s. and |Bk |p = κpk .
For k = 1 formula (12) was recently obtained in [9].
For p = 1, the right-hand side of (12) is proportional to the k-th intrinsic volume of E
(see (2)), which implies the following result (for details see Subsection 3.2).
RANDOM AFFINE SIMPLEXES
5
Corollary 1.8. For X0 , . . . , Xk uniformly distributed in any non-degenerate ellipsoid E ⊂ Rd
we have
!2
κk+1
1 ((d + 1)!)k+1
d+1
Vk (E).
E | conv(X0 , . . . , Xk )| = k
2 ((d + 1)(k + 1))! κ(d+1)(k+1)
Very recently, for X0 , . . . , Xk uniformly distributed in the unit ball Bd , a formula for
the distribution of | conv(X0 , . . . , Xk )| has been derived in [7]. For a random variable η and
α1 , α2 > 0 we write η ∼ B(α1 , α2 ) to denote that η has a Beta distribution with parameters
α1 , α2 and the density
Γ(α1 + α2 ) α1 −1
t
(1 − t)α2 −1 ,
Γ(α1 ) Γ(α2 )
t ∈ (0, 1).
It was shown in [7] that for X0 , . . . , Xk uniformly distributed in Bd ,
d
(k!)2 η(1 − η)k | conv(X0 , . . . , Xk )|2 = (1 − η ′ )k η1 · · · ηk ,
(13)
where η, η ′ , η1 , . . . , ηk are independent random variables independent with X0 , . . . , Xk such
that
d−k+i k−i
kd
d
′
, ηi ∼ B
+ 1,
,
+1 .
η, η ∼ B
2
2
2
2
Multiplying both sides of (13) by |Pξ E|2 /κ2k and applying Theorem 1.1 and Corollary 1.4
(for details see Subsection 3.1) leads to the following generalization of (13).
Theorem 1.9. For X0 , . . . , Xk uniformly distributed in any non-degenerate ellipsoid E ⊂ Rd
we have
d
′ k
2
(k!)2 η (1 − η)k | conv(X0 , . . . , Xk )|2 = κ−2
k (1 − η ) η1 · · · ηk |Pξ E|
d
= (1 − η ′ )k η1 · · · ηk
!
det G⊤
λ Gλ
,
det (G⊤ G)
where the matrices G and Gλ are defined in Corollary 1.4 and λ1 , . . . , λd denoting the length
of semi-axes of E.
Taking k = 1 yields the distribution of the distance between two random points in E.
Corollary 1.10. Under the assumptions of Theorem 1.9 we have
2 2
λ1 N1 + · · · + λ2d Nd2
2 d
′
η(1 − η) · |X0 − X1 | = (1 − η ) η1
,
N12 + · · · + Nd2
where N1 , . . . , Nd are i.i.d. standard Gaussian variables.
6
FRIEDRICH GÖTZE, ANNA GUSAKOVA, AND DMITRY ZAPOROZHETS
1.4. Integral geometry formulas. Recall that Gd,k and Ad,k denote the linear and affine
Grassmannians defined in Subsection 1.1.
For an arbitrary convex compact body K, p ∈ Z+ , and k = 1 it is possible to express (9)
in terms of the lengths of the one-dimensional sections of K (see [3] and [13]):
Z
Z
2dκd
p
|K ∩ E|p+d+1 µd,1 (dE).
|x0 − x1 | dx0 dx1 =
(d + p) (d + p + 1)
Ad,1
K2
This formula does not extend to k > 1. For ellipsoids K = E this is possible (for details see
Subsection 4.1).
Theorem 1.11. For any non-degenerate ellipsoid E ⊂ Rd , k ∈ {0, 1, . . . , d}, and any real
number p > −d + k − 1 we have
Z
Z
κk+1
κk(d+p)+k bd,k
d+p
p
|E ∩ E|p+d+1 µd,k (dE).
| conv(x0 , . . . , xk )| dx0 . . . dxk = p+d+1
κ
b
κk
(k+1)(d+p) d+p,k
Ad,k
E k+1
(14)
Comparing this theorem with Theorem 1.7 readily gives the following connection between the volumes of k-dimensional cross-sections and projections of ellipsoids.
Theorem 1.12. Under the assumptions of Theorem 1.11 we have
Z
Z
κk(d+p)+k
κk+1
p+d+1
k+1
d
|E ∩ E|
µd,k (dE) = |E|
|PL E|p νd,k (dL).
d+1 κ
κk
k(d+p)+d
Ad,k
Gd,k
For p = 0, we obtain the following integral formula.
Corollary 1.13. Under the assumptions of Theorem 1.11 we have
Z
κd(k+1)
κd+1
k
|E ∩ E|d+1 µd,k (dE) = k+1
|E|k+1.
κd κk(d+1)
(15)
Ad,k
This result may be regarded as an affine version of the following integral formula of
Furstenberg and Tzkoni (see [5]):
Z
κd
|E ∩ L|d νd,k (dL) = kk |E|k .
κd
Gd,k
Our next theorem generalizes this formula in the same way as Theorem 1.11 generalizes (15).
Theorem 1.14. For any non-degenerate ellipsoid E ⊂ Rd , k ∈ {0, 1, . . . , d}, and any real
number p > −d + k we have
Z
Z
κkd+p bd,k
p
| conv(0, x1 , . . . , xk )| dx1 . . . dxk = p+d
|E ∩ L|p+d νd,k (dL).
(16)
κk bd+p,k
Ek
Gd,k
RANDOM AFFINE SIMPLEXES
7
In probabilistic language it may be formulated as
κkd+p bd,k
E | conv(0, X1 , . . . , Xk )| = p+d
E |E ∩ ξ|p+d,
κk bd+p,k
p
where X1 , . . . , Xk are i.i.d. random vectors uniformly distributed in E and ξ is a uniformly
chosen random k-dimensional linear subspace in Rd .
2. Proofs: Part I
2.1. Blaschke-Petkantschin formula. In our calculations we will need to integrate some
non-negative measurable function h of k-tuples of points in Rd . To this end, we integrate
first over the k-tuples of points in a fixed k-dimensional linear subspace L, with respect to
the product measure λkL , and then integrate over Gd,k , with respect to νd,k . The corresponding transformation formula is known as the linear Blaschke-Petkantschin formula (see [18,
Theorem 7.2.1]):
Z
h(x1 , . . . , xk ) dx1 . . . dxk =
(17)
(Rd )k
bd,k
Z Z
h(x1 , . . . , xk ) | conv(0, x1 , . . . , xk )|d−k λL (dx1 ) . . . λL (dxk ) νd,k (dL),
Gd,k Lk
where bd,k is defined in (11).
A similar affine version (see [18, Theorem 7.2.7]) may be stated as follows:
Z
h(x0 , . . . , xk ) dx0 . . . dxk =
(18)
(Rd )k+1
bd,k
Z
Z
h(x0 , . . . , xk ) | conv(x0 , . . . , xk )|d−k λE (dx0 ) . . . λE (dxk ) µd,k (dE).
Ad,k E k+1
2.2. Proof of Proposition 1.3. To avoid trivialities we assume that dim L = k. Let
e1 , . . . , ek ∈ Rd be some orthonormal basis in L. Let OL and X denote d × k matrices whose
columns are e1 , . . . , ek and x1 , . . . , xk respectively. It is easy to check that OL OL⊤ is a d × d
matrix corresponding to the orthogonal projection operator PL . Thus,
OL OL⊤ X = X.
(19)
Recall that E is defined by (4). It is known (see, e.g., [19, Appendix H]) that the
orthogonal projection PL E is an ellipsoid in L and
h
i1/2
|PL E| = κk det OL⊤ HOL
,
(20)
where
H := A⊤ A.
8
FRIEDRICH GÖTZE, ANNA GUSAKOVA, AND DMITRY ZAPOROZHETS
A well-known formula for the volume of a k-dimensional simplex states that for any x1 , . . . , xk ∈
Rd ,
i1/2
1h
det X ⊤ X
.
(21)
| conv(0, x1 , . . . , xk )| =
k!
Therefore,
h
i1/2 h
i1/2
k! | conv(0, Ax1 , . . . , Axk )| = det (AX)⊤ AX
= det X ⊤ HX
.
Applying (19) produces
det X ⊤ HX = det X ⊤ OL OL⊤ HOL OL⊤ X = det OL⊤ HOL det X ⊤ OL det OL⊤ X
= det OL⊤ HOL det X ⊤ OL OL⊤ X = det OL⊤ HOL det X ⊤ X ,
which together with (20) and (21) finishes the proof.
2.3. Proof of Theorem 1.1. Denote by f (x0 , . . . , xk ) the joint density of (X0 , . . . , Xk ).
Let
Z
ϕA (t) :=
exp it log | conv(Ax0 , . . . , Axk )| f (x0 , . . . , xk ) dx0 . . . dxk
(Rd )k+1
=
Z
(Rd )k+1
exp it log | conv(0, A(x1 − x0 ), . . . , A(xk − x0 ))| f (x0 , . . . , xk ) dx0 . . . dxk
be a characteristic function of log | conv(AX0 , . . . , AXk )|. In particular, denoting by I the
identity matrix, we obtain that ϕI (t) is a characteristic function of log | conv(X0 , . . . , Xk )|.
Substituting y0 = x0 and yi = xi − x0 for 1 ≤ i ≤ k leads to
Z
ϕA (t) =
exp it log | conv(0, Ay1 , . . . , Ayk )| f (y0 , y1 + y0 . . . , yk + y0 ) dy0 . . . dyk
(Rd )k+1
=
Z
(Rd )k
exp it log | conv(0, Ay1 , . . . , Ayk )| g(y1 , . . . , yk ) dy1 . . . dyk ,
where
g(y1 , . . . , yk ) :=
Z
f (y0 , y1 + y0 . . . , yk + y0 ) dy0 .
Rd
Using the linear Blaschke-Petkantschin formula (see (17)) with
gives
ϕA (t) =
h(y1 , . . . , yk ) := exp it log | conv(0, Ay1 , . . . , Ayk )| g(y1 , . . . , yk )
Z Z
Gd,k Lk
exp it log | conv(0, Ay1 , . . . , Ayk )| g(y1 , . . . , yk )×
| conv(0, y1 , . . . , yk )|d−k λL (dy1 ) . . . λL (dyk ) νd,k (dL). (22)
RANDOM AFFINE SIMPLEXES
9
Applying Proposition 1.3 to (22) gives
Z
Z
|PL E|
exp it log | conv(0, y1 , . . . , yk )| g(y1 , . . . , yk )
ϕA (t) =
exp it log
κk
Gd,k
Lk
× | conv(0, y1 , . . . , yk )|d−k λL (dy1 ) . . . λL (dyk ) νd,k (dL).
Since f is spherically invariant, the function
Z
hA (t) := exp it log | conv(0, y1 , . . . , yk )| g(y1 , . . . , yk )
Lk
× | conv(0, y1 , . . . , yk )|d−k λL (dy1 ) . . . λL (dyk )
does not depend on the choice of L. Indeed, consider any L′ ∈ Gd,k . There exists an
orthogonal matrix U such that L = UL′ . Substituting yi = Uzi gives
Z
hA (t) =
exp it log | conv(0, Uz1 , . . . , Uzk )| g(Uz1 , . . . , Uzk )
L′ k
× | conv(0, Uz1 , . . . , Uzk )|d−k λ′L (dz1 ) . . . λ′L (dzk ).
Now the claim follows from
| conv(0, Uz1 , . . . , Uzk )| = | conv(0, z1 , . . . , zk )|
and
g(Uz1 , . . . , Uzk ) =
Z
f (y0 , Uz1 + y0 . . . , Uzk + y0 ) dy0
Z
f (Uy0 , Uz1 + Uy0 . . . , Uzk + Uy0 ) dy0
Z
f (y0 , z1 + y0 . . . , zk + y0 ) dy0
Rd
=
Rd
=
Rd
= g(z1 , . . . , zk ),
where at the second step we did a variable change y0 → Uy0 and at the third step we used
the spherical symmetry of f .
Thus hA (t) does not depend on the choice of L, which implies
|Pξ E|
ϕA (t) = hA (t) E exp it log
.
κk
In particular,
ϕI (t) = hA (t).
10
FRIEDRICH GÖTZE, ANNA GUSAKOVA, AND DMITRY ZAPOROZHETS
Comparing the last two equalities and applying the characteristic function uniqueness theorem, we arrive at
d
log | conv(AX0 , . . . , AXk )| = log
|Pξ E|
+ log | conv(X0 , . . . , Xk )|,
κk
and the theorem follows.
2.4. Proof of Corollary 1.4. Denote by G1 , . . . , Gk ∈ Rd the columns of the matrix G.
Hence, AG1 , . . . , AGk ∈ Rd are the columns of the matrix AG. Using Proposition 1.3 with
xi = Gi and applying (21) to G and AG gives
h
or
⊤
⊤
det G A AG
i1/2
i1/2
|Pη E| h
⊤
· det G G
,
=
κk
det G⊤ A⊤ AG
det G⊤ G
!1/2
=
|Pη E|
,
κk
where η is the linear hull of 0, G1 , . . . , Gk . Since G1 , . . . , Gk are i.i.d. standard Gaussian
d
vectors, η is uniformly distributed in Gd,k with respect to νd,k , which implies η = ξ, and the
corollary follows.
3. Proofs: Part II
3.1. Proofs of Theorem 1.7 and Theorem 1.9. For any non-degenerate ellipsoid E there
exist a unique symmetric positive-definite d × d matrix A such that
E = ABd = x ∈ Rd : kA−1 xk ≤ 1 = x ∈ Rd : x⊤ A−2 x ≤ 1 .
Since X0 , . . . , Xk are i.i.d. random vectors uniformly distributed in E, we have that A−1 X0 ,
. . . , A−1 Xk are i.i.d. random vectors uniformly distributed in Bd . It follows from Theorem 1.1
that
d
d
| conv(X0 , . . . , Xk )| = | conv(AA−1 X0 , . . . , AA−1 Xk )| = | conv(A−1 X0 , . . . , A−1 Xk )|
Taking the p-th moment and applying (10) implies Theorem 1.7.
Now apply (13) to A−1 X0 , . . . , A−1 Xk :
d
(k!)2 η(1 − η)k | conv(A−1 X0 , . . . , A−1 Xk )|2 = (1 − η ′ )k η1 · · · ηk .
Multiplying by
|Pξ E|
κpk
and applying (23) implies Theorem 1.9.
|Pξ E|
.
κpk
(23)
RANDOM AFFINE SIMPLEXES
11
3.2. Proof of Corollary 1.8. From Kubota’s formula (see (2)) and Theorem 1.7 we have
E | conv(X0 , . . . , Xk )| = αd,k Vk (E),
where
κk+1
bd,k κd−k
d+1 κk(d+1)+d
.
d
k+1 κ
κd
(k+1)(d+1) bd+1,k k κd
(see (11)) and κp (see (1)) we obtain
αd,k :=
From the definition of bd,k
κk+1
d+1 κk(d+1)+d (d + 1 − k)! κd−k+1 κd−k
κ(k+1)(d+1)
(d + 1)! κd+1
κd
κk+1
d
!k+1
Γ 12 d + 1
(d + 1 − k)!
= k/2
π (d + 1)! Γ 21 (d + 1) + 1
Γ 21 (d + 1) + 1
Γ 21 d + 1
Γ 21 (k + 1)(d + 1) + 1
.
×
Γ 21 ((k + 1)d + k) + 1 Γ 21 (d − k + 1) + 1 Γ 12 (d − k) + 1
αd,k =
Using Legendre’s duplication formula for the Gamma function
1
= 21−2z π 1/2 Γ (2z) ,
Γ (z) Γ z +
2
the recursion Γ (1 + z) = z Γ (z), and the fact that k, d ∈ Z we obtain
!k+1
Γ 21 d + 1
Γ 12 d + 21 Γ 12 d + 1
(d − k)! Γ 21 (k + 1)(d + 1) + 1
αd,k = k/2
π d! Γ 12 ((k + 1)d + k) + 1 Γ 21 (d − k) + 21 Γ 12 (d − k) + 1
Γ 12 (d + 1) + 1
!k+1
Γ 12 (k + 1)(d + 1) + 1
Γ 21 d + 1
1
= √ k
1
Γ 12 (d + 1) + 1
(2 π) Γ 2 ((k + 1)d + k) + 1
!2
k+1
κk+1
Γ 12 d + 1 Γ 12 d + 1 + 12
1
d+1
= √ k
1
1
1
κ
Γ
(kd
+
d
+
k)
+
1
Γ
(kd
+
k
+
d)
+
1
+
(d+1)(k+1)
(2 π)
2
2
2
!2
κk+1
1 ((d + 1)!)k+1
d+1
= k
.
2 ((d + 1)(k + 1))! κ(d+1)(k+1)
4. Proofs: Part III
4.1. Proof of Theorem 1.11. Let
Z
J :=
| conv(x0 , . . . , xk )|p dx0 . . . dxk =
E k+1
Z
| conv(x0 , . . . , xk )|p
(Rd )k+1
Using the affine Blaschke-Petkantschin formula (see (18)) with
h(x0 , . . . , xk ) := | conv(x0 , . . . , xk )|p
k
Y
i=0
1E (xi )
k
Y
i=0
1E (xi ) dx0 . . . dxk .
12
FRIEDRICH GÖTZE, ANNA GUSAKOVA, AND DMITRY ZAPOROZHETS
yields
J = bd,k
Z
Z
| conv(x0 , . . . , xk )|
p+d−k
= bd,k
Z
1E (xi ) λE (dx0 ) . . . λE (dxk ) µd,k (dE)
i=0
Ad,k E k+1
Z
k
Y
| conv(x0 , . . . , xk )|p+d−k λE (dx0 ) . . . λE (dxk ) µd,k (dE).
Ad,k (E∩E)k+1
Now fix E ∈ Ad,k . Applying Theorem 1.7 to the ellipsoid E ∩ E gives
1
|E ∩ E|k+1
Z
| conv(x0 , . . . , xk )|p+d−k λE (dx0 ) . . . λE (dxk )
(E∩E)k+1
=
κk+1
d+p
κp+d+1
k
κk(d+p)+k
1
|E ∩ E|p+d−k ,
κ(k+1)(d+p) bd+p,k
which leads to
J=
κk+1
d+p
κp+d+1
k
κk(d+p)+k bd,k
κ(k+1)(d+p) bd+p,k
Z
|E ∩ E|p+d+1 µd,k (dE).
Ad,k
4.2. Proof of Theorem 1.14. The proof is similar to the previous one. Let
J :=
Z
p
| conv(0, x1 , . . . , xk )| dx1 . . . dxk =
Ek
Z
| conv(0, x1 , . . . , xk )|
(Rd )k
p
k
Y
1E (xi ) dx1 . . . dxk .
i=1
Using the linear Blaschke-Petkantschin formula (see (17)) with
h(x1 , . . . , xk ) := | conv(0, x1 , . . . , xk )|
p
k
Y
1E (xi )
i=1
gives
J = bd,k
Z Z
| conv(0, x1 , . . . , xk )|
Gd,k Lk
= bd,k
Z
Z
Gd,k (L∩E)k
p+d−k
k
Y
1E (xi ) λL (dx1 ) . . . λL (dxk ) νd,k (dL)
i=1
| conv(0, x1 , . . . , xk )|p+d−k λL (dx1 ) . . . λL (dxk ) νd,k (dL). (24)
RANDOM AFFINE SIMPLEXES
13
Fix L ∈ Gd,k . Since E ∩ L is an ellipsoid, there exists a linear transform AL : L → Rk such
that AL (E ∩ L) = Bk . Applying the coordinate transform xi = AL yi , i = 1, 2, . . . , k, we get
Z
| conv(0, x1 , . . . , xk )|p+d−k λL (dx1 ) . . . λL (dxk )
(L∩E)k
=
|E ∩ L|p+d
κp+d
k
Z
| conv(0, y1 , . . . , yk )|p+d−k λL (dy1 ) . . . λL (dyk ). (25)
(Bk )k
It is known (see, e.g., [18, Theorem 8.2.2]) that
Z
k
Y
p+d−k
−p−d+k k
| conv(0, y1 , . . . , yk )|
λL (dy1 ) . . . λL (dyk ) = (k!)
κd+p
i=1
(Bk )k
(k + 1 − i) κk+1−i
.
(d + p + 1 − i) κd+p+1−i
(26)
Substituting (26) and (25) into (24) finishes the proof.
4.3. Acknowledgements. The authors are grateful to Günter Last and Daniel Hug for
helpful discussions and suggestions which improved this paper.
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Friedrich Götze: Faculty of Mathematics, Bielefeld University, P. O. Box 10 01 31,
33501 Bielefeld, Germany
E-mail address: goetze@math.uni-bielefeld.de
Anna Gusakova: Faculty of Mathematics, Bielefeld University, P. O. Box 10 01 31,
33501 Bielefeld, Germany
E-mail address: agusakov@math.uni-bielefeld.de
Dmitry Zaporozhets: St. Petersburg Department of Steklov Mathematical Institute,
Fontanka 27, 191023 St. Petersburg, Russia
E-mail address: zap1979@gmail.com