arXiv:1711.06578v1 [math.PR] 17 Nov 2017
FRIEDRICH G ¨OTZE, 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-singulard×d
matrix. We show that thek-dimensional volume of the convex hull of affinely transformed
Xi’s satisfies
|conv(AX0, . . . , AXk)| d
= |PξE|
κk · |
conv(X0, . . . , Xk)|,
where E := {x ∈ Rd : x⊤(A⊤A)−1x ≤ 1} is an ellipsoid, Pξ denotes the orthogonal
projection to a random uniformly chosen k-dimensional linear subspace ξ independent of
X0, . . . , Xk, andκk is the volume of the unit k-dimensional ball.
We express |PξE| in terms of Gaussian random matrices. The important special case k= 1 corresponds to the distance between two random points:
|AX0−AX1|=d
s λ2
1N12+· · ·+λ2dNd2 N2
1+· · ·+Nd2
· |X0−X1|,
whereN1, . . . , Ndare i.i.d. standard Gaussian variables independent ofX0, X1andλ1, . . . , λd
are the singular values ofA.
As an application, we derive the following integral geometry formula for ellipsoids:
κkd+1 κdk+1
κk(d+p)+k κk(d+p)+d
Z
Ad,k
|E ∩E|p+d+1µd,k(dE) =|E|k+1 Z
Gd,k
|PLE|pνd,k(dL),
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 casep= 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 follow-ing [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 k -dimensional linear subspaces, and intersections with k-dimensional affine subspaces, where
k ∈ {0, . . . , d}. In this case, | · |stands for k-dimensional volume.
2010Mathematics 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”.
The unit ball in Rk is denoted by
Bk. For p >0 we write
κp :=
πp/2
Γ p2 + 1, (1)
where for an integer k we have κk =|Bk|.
Fork ∈ {0, . . . , d}, the linear (resp., affine) Grassmannian ofk-dimensional linear (resp., affine) subspaces ofRdis denoted byG
d,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∩Bd6=∅ =κd−k.
A compact convex subsetK of Rd with non-empty interior is called a convex body. We define the intrinsic volumes of K by Kubota’s formula:
Vk(K) =
d k
κd
κkκd−k
Z
Gd,k
|PLK|νd,k(dL), (2)
where PLK 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? Fork=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)−1x≤1 . (4)
Then we have
|conv(AX0, . . . , AXk)| d
= |PξE|
κk · |
conv(X0, . . . , Xk)|, (5)
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|PξE| is proportional to Vk(E). Thus, taking
Corollary 1.2. Under the assumptions of Theorem 1.1 we have
E|conv(AX0, . . . , AXk)|=
d k
−1
κkκd−k
κd
Vk(E)E|conv(X0, . . . , Xk)|. (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
|conv(0, Ax1, . . . , Axk)|= |
PLE|
κk · |
conv(0,x1, . . . ,xk)|. (7)
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
|PξE|
κk d
= det G
⊤A⊤AG
det G⊤G
!1/2
d
= det G
⊤
λGλ
det G⊤G !1/2
, (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 λiNij, 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
|conv(AX0, . . . , AXk)| d
= det G
⊤A⊤AG
det G⊤G
!1/2
|conv(X0, . . . , Xk)|
d
= det G
⊤
λGλ
det G⊤G !1/2
|conv(X0, . . . , Xk)|.
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
|AX0−AX1|
d
= s
λ2
1N12+· · ·+λ2dNd2
N2
1 +· · ·+Nd2
· |X0−X1|,
These results will be used in the next subsection to study integral geometry problems for ellipsoids.
1.3. Random points in ellipsoids. Now suppose thatX0, . . . , Xkare uniformly distributed
in some convex set K ⊂ Rd with non-empty interior. A classical problem of stochastic ge-ometry is to find the distribution of (3) starting with its moments
E|conv(X0, . . . , Xk)|p =
1 |K|k+1
Z
Kk+1
|conv(x0, . . . ,xk)|pdx0. . .dxk. (9)
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 arbitrarydandk = 1, there is an electromagnetic interpretation of (9) (see [8]): a transmitterX0 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
E|conv(X0, . . . , Xk)|p =
κkd++1p κkd+1
κk(d+p)+d
κ(k+1)(d+p)
bd,k
bd+p,k
, (10)
where κk are are defined in (1) and for any real number q > k−1 we write
bq,k := (k!)q−k−1(q−k+ 1) · · · q·
κq−k+1· · ·κq
κ1· · ·κk
. (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 withX0, . . . , Xk
Theorem 1.7. ForX0, . . . , Xk uniformly distributed in any non-degenerate ellipsoidE ⊂Rd
and any real number p >−d+k−1 we have
E|conv(X0, . . . , Xk)|p =
κkd++1p κkd+1
κk(d+p)+d
κ(k+1)(d+p)
bd,k
bd+p,k
E|PξE|p
κpk . (12)
Note that (12) is indeed a generalization of (10) since PξBd =Bk a.s. and |Bk|p =κpk.
For k= 1 formula (12) was recently obtained in [9].
Corollary 1.8. ForX0, . . . , Xkuniformly distributed in any non-degenerate ellipsoidE ⊂Rd
we have
E|conv(X0, . . . , Xk)|=
1 2k
((d+ 1)!)k+1 ((d+ 1)(k+ 1))!
κkd+1+1 κ(d+1)(k+1)
!2
Vk(E).
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) Γ(α1) Γ(α2)
tα1−1(1−t)α2−1, t∈(0,1).
It was shown in [7] that for X0, . . . , Xk uniformly distributed in Bd,
(k!)2η(1−η)k|conv(X0, . . . , Xk)|2 = (1d −η′)kη1· · ·ηk, (13)
where η, η′, η
1, . . . , ηk are independent random variables independent with X0, . . . , Xk such
that
η, η′ ∼B
d
2+ 1,
kd
2
, ηi ∼B
d−k+i
2 ,
k−i
2 + 1
.
Multiplying both sides of (13) by |PξE|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. ForX0, . . . , Xk uniformly distributed in any non-degenerate ellipsoidE ⊂Rd
we have
(k!)2η(1−η)k|conv(X
0, . . . , Xk)|2 d
=κ−k2(1−η′)kη
1· · ·ηk|PξE|2
d
= (1−η′)kη
1· · ·ηk
det G⊤
λGλ
det (G⊤G)
!
,
where the matrices Gand Gλ are defined in Corollary 1.4 andλ1, . . . , λddenoting 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
η(1−η) · |X0−X1|2
d
= (1−η′)η1
λ2
1N12 +· · ·+λ2dNd2
N2
1 +· · ·+Nd2
,
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 bodyK,p∈Z+, andk = 1 it is possible to express (9) in terms of the lengths of the one-dimensional sections of K (see [3] and [13]):
Z
This formula does not extend to k >1. For ellipsoids K =E this is possible (for details see Subsection 4.1).
Comparing this theorem with Theorem 1.7 readily gives the following connection be-tween the volumes ofk-dimensional cross-sections and projections of ellipsoids.
Theorem 1.12. Under the assumptions of Theorem 1.11 we have
κkd+1
For p= 0, we obtain the following integral formula.
Corollary 1.13. Under the assumptions of Theorem 1.11 we have
Z
This result may be regarded as an affine version of the following integral formula of Furstenberg and Tzkoni (see [5]):
Z
Our next theorem generalizes this formula in the same way as Theorem 1.11 generalizes (15).
In probabilistic language it may be formulated as
E|conv(0, X1, . . . , Xk)|p =
κk d+p
κpk+d bd,k
bd+p,k E|E ∩
ξ|p+d,
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 λk
L, and then integrate overGd,k, with respect toνd,k. The
correspond-ing transformation formula is known as the linear Blaschke-Petkantschin formula (see [18, Theorem 7.2.1]):
Z
(Rd)k
h(x1, . . . ,xk) dx1. . .dxk= (17)
bd,k
Z
Gd,k Z
Lk
h(x1, . . . ,xk)|conv(0,x1, . . . ,xk)|d−kλL(dx1). . . λL(dxk)νd,k(dL),
where bd,k is defined in (11).
A similar affine version (see [18, Theorem 7.2.7]) may be stated as follows: Z
(Rd)k+1
h(x0, . . . ,xk) dx0. . .dxk = (18)
bd,k
Z
Ad,k Z
Ek+1
h(x0, . . . ,xk)|conv(x0, . . . ,xk)|d−kλE(dx0). . . λE(dxk)µd,k(dE).
2.2. Proof of Proposition 1.3. To avoid trivialities we assume that dimL = k. Let
e1, . . . ,ek ∈Rd be some orthonormal basis inL. LetOLand X denoted×k matrices whose
columns are e1, . . . ,ek and x1, . . . ,xk respectively. It is easy to check that OLOL⊤ is a d×d
matrix corresponding to the orthogonal projection operatorPL. Thus,
OLO⊤LX =X. (19)
Recall that E is defined by (4). It is known (see, e.g., [19, Appendix H]) that the orthogonal projection PLE is an ellipsoid in Land
|PLE|=κk
h
det O⊤LHOL
i1/2
, (20)
where
A well-known formula for the volume of ak-dimensional simplex states that for anyx1, . . . ,xk ∈
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).
Using the linear Blaschke-Petkantschin formula (see (17)) with
Applying Proposition 1.3 to (22) gives
ϕA(t) =
Z
Gd,k exp
itlog|PLE|
κk
Z
Lk
exp itlog|conv(0,y1, . . . ,yk)|
g(y1, . . . ,yk)
× |conv(0,y1, . . . ,yk)|d−kλL(dy1). . . λL(dyk)νd,k(dL).
Since f is spherically invariant, the function
hA(t) :=
Z
Lk
exp itlog|conv(0,y1, . . . ,yk)|
g(y1, . . . ,yk)
× |conv(0,y1, . . . ,yk)|d−kλ
L(dy1). . . λL(dyk)
does not depend on the choice of L. Indeed, consider any L′ ∈ G
d,k. There exists an
orthogonal matrix U such thatL=UL′. Substituting y
i =Uzi gives
hA(t) =
Z
L′k
exp itlog|conv(0, Uz1, . . . , Uzk)|
g(Uz1, . . . , Uzk)
× |conv(0, Uz1, . . . , Uzk)|d−kλ′L(dz1). . . λ′L(dzk).
Now the claim follows from
|conv(0, Uz1, . . . , Uzk)|=|conv(0,z1, . . . ,zk)|
and
g(Uz1, . . . , Uzk) = Z
Rd
f(y0, Uz1+y0. . . , Uzk+y0) dy0
= Z
Rd
f(Uy0, Uz1+Uy0. . . , Uzk+Uy0) dy0
= Z
Rd
f(y0,z1+y0. . . ,zk+y0) dy0
=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
ϕA(t) =hA(t)E exp
itlog |PξE|
κk
.
In particular,
Comparing the last two equalities and applying the characteristic function uniqueness theo-rem, we arrive at
log|conv(AX0, . . . , AXk)| d
= log|PξE|
κk
+ log|conv(X0, . . . , Xk)|,
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 matrixAG. Using Proposition 1.3 with
xi =Gi and applying (21) to Gand AGgives
h
det G⊤A⊤AGi1/2
= |PηE|
κk ·
h
det G⊤Gi1/2
,
or
det G⊤A⊤AG det G⊤G
!1/2
= |PηE|
κk
,
where η is the linear hull of 0, G1, . . . , Gk. Since G1, . . . , Gk are i.i.d. standard Gaussian
vectors, η is uniformly distributed in Gd,k with respect to νd,k, which implies η d
=ξ, and the corollary follows.
3. Proofs: Part II
3.1. Proofs of Theorem 1.7 and Theorem 1.9. For any non-degenerate ellipsoidE there exist a unique symmetric positive-definite d×d matrix A such that
E =ABd=
x∈Rd:kA−1xk ≤1 =
x∈Rd :x⊤A−2x≤1 .
Since X0, . . . , Xk are i.i.d. random vectors uniformly distributed in E, we have thatA−1X0,
. . . , A−1X
kare i.i.d. random vectors uniformly distributed inBd. It follows from Theorem 1.1
that
|conv(X0, . . . , Xk)| d
=|conv(AA−1X0, . . . , AA−1Xk)| d
=|conv(A−1X0, . . . , A−1Xk)||
PξE|
κpk .
(23) Taking the p-th moment and applying (10) implies Theorem 1.7.
Now apply (13) to A−1X
0, . . . , A−1Xk:
(k!)2η(1−η)k|conv(A−1X0, . . . , A−1Xk)|2 d
= (1−η′)kη1· · ·ηk.
Multiplying by |PξE|
3.2. Proof of Corollary 1.8. From Kubota’s formula (see (2)) and Theorem 1.7 we have Using Legendre’s duplication formula for the Gamma function
Γ (z) Γ
4.1. Proof of Theorem 1.11. Let
J :=
Using the affine Blaschke-Petkantschin formula (see (18)) with
h(x0, . . . ,xk) :=|conv(x0, . . . ,xk)|p k
Y
i=0
yields
4.2. Proof of Theorem 1.14. The proof is similar to the previous one. Let
J :=
Using the linear Blaschke-Petkantschin formula (see (17)) with
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 =ALyi, i= 1,2, . . . , k, we get
Z
(L∩E)k
|conv(0,x1, . . . ,xk)|p+d−kλL(dx1). . . λL(dxk)
= |E ∩L|
p+d
κpk+d
Z
(Bk)k
|conv(0,y1, . . . ,yk)|p+d−kλL(dy1). . . λL(dyk). (25)
It is known (see, e.g., [18, Theorem 8.2.2]) that
Z
(Bk)k
|conv(0,y1, . . . ,yk)|p+d−kλL(dy1). . . λL(dyk) = (k!)−p−d+kκkd+p k
Y
i=1
(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 G¨unter Last and Daniel Hug for helpful discussions and suggestions which improved this paper.
References
[1] U. B¨asel. Random chords and point distances in regular polygons. Acta Math. Univ. Comenianae, 83(1):1–18, 2014.
[2] ´E. Borel. Principes et formules classiques du calcul des probabilit´es. 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. T¨ale. 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.
[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
appli-cations, volume 1 ofStatistical 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. Santal´o. 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 G¨otze: 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
