# A Blaschke–Petkantschin formula for linear and affine subspaces with
application to intersection probabilities

Emil Dare111Aarhus University, Department of Mathematics, Aarhus C, Denmark.
Email<EMAIL_ADDRESS>, Markus Kiderlen222Aarhus University, Department of
Mathematics, Aarhus C, Denmark. Email<EMAIL_ADDRESS>and Christoph
Thäle333 Ruhr University Bochum, Faculty of Mathematics, Bochum, Germany.
Email<EMAIL_ADDRESS>
###### Abstract

Consider a uniformly distributed random linear subspace $L$ and a
stochastically independent random affine subspace $E$ in $\mathbb{R}^{n}$,
both of fixed dimension. For a natural class of distributions for $E$ we show
that the intersection $L\cap E$ admits a density with respect to the invariant
measure. This density depends only on the distance $d(o,E\cap L)$ of $L\cap E$
to the origin and is derived explicitly. It can be written as the product of a
power of $d(o,E\cap L)$ and a part involving an incomplete beta integral.
Choosing $E$ uniformly among all affine subspaces of fixed dimension hitting
the unit ball, we derive an explicit density for the random variable
$d(o,E\cap L)$ and study the behavior of the probability that $E\cap L$ hits
the unit ball in high dimensions. Lastly, we show that our result can be
extended to the setting where $E$ is tangent to the unit sphere, in which case
we again derive the density for $d(o,E\cap L)$. Our probabilistic results are
derived by means of a new integral-geometric transformation formula of
Blaschke–Petkantschin type.

Keywords. Blaschke–Petkantschin formula, integral geometry, intersection
probability, stochastic geometry.
MSC. 52A22, 53C65, 60D05.

##  1 Introduction and motivation

Fix dimension parameters $n\geq 2$, $q\in\\{1,\ldots,n-1\\}$ and
$\gamma\in\\{0,\ldots,q-1\\}$. Let $L_{1}$ be a $q$-dimensional random linear
subspace and $L_{2}$ be an $(n-q+\gamma)$-dimensional random linear subspace
of $\mathbb{R}^{n}$. We assume that both subspaces are stochastically
independent and that $L_{1}$ and $L_{2}$ are selected according to the uniform
distribution on the Grassmannian $G(n,k)$ of all $k$-dimensional linear
subspaces of $\mathbb{R}^{n}$, with $k=q$ and $k=n-q+\gamma$, respectively. In
other words, we use the normalized rotation invariant measures $\nu_{k}$,
$k=q$ and $k=n-q+\gamma$, on these spaces as our underlying probability
measures; these and further concepts will formally be introduced in Section 2.
The intersection $L_{1}\cap L_{2}$ is almost surely a random subspace of
$\mathbb{R}^{n}$ of dimension $\gamma$ and its distribution is known to be the
uniform distribution on the space $G(n,\gamma)$, see Figure 1.1.

Figure 1.1: Illustration in the case $n=2$, $q=1$, $\gamma=0$. From left to
right: Intersection of two linear subspaces; intersection of two affine
subspaces hitting the unit ball; intersection of a linear with an affine
subspace hitting the unit ball; intersection of a linear subspace with an
affine subspace tangent to the unit sphere.

Let us now change the set-up and let $E_{1}$ be a $q$-dimensional random
affine subspace and $E_{2}$ be another $(n-q+\gamma)$-dimensional random
affine subspace of $\mathbb{R}^{n}$. Since the motion invariant measure
$\mu_{k}$ on $A(n,k)$, $k\in\\{0,\ldots,n-1\\}$, is not finite, we restrict
attention to the set

$[B^{n}]_{k}=\\{E\in A(n,k):E\cap B^{n}\neq\emptyset\\}$

of $k$-dimensional affine subspaces hitting the unit ball $B^{n}$ of
$\mathbb{R}^{n}$. We thus take $E_{1}$ and $E_{2}$ as random affine subspaces
distributed according to the normalized measures $\mu_{q}$ and
$\mu_{n-q+\gamma}$ restricted to $[B^{n}]_{q}$ and $[B^{n}]_{n-q+\gamma}$,
respectively. Assuming that $E_{1}$ and $E_{2}$ are stochastically
independent, the intersection $E_{1}\cap E_{2}$ is almost surely a random
affine subspace of $\mathbb{R}^{n}$ with dimension $\gamma$. However, the
intersection of $E_{1}\cap E_{2}$ with the unit ball $B^{n}$ may or may not be
empty with strictly positive probability, see Figure 1.1. Already this basic
observation shows that – in contrast to the case of random linear subspace
discussed above – the distribution of $E_{1}\cap E_{2}$ cannot coincide with
the normalized motion-invariant measure on $[B^{n}]_{\gamma}$. More precisely,
since the distance of the intersection of $E_{1}$ and $E_{2}$ to the origin
$o\in\mathbb{R}^{n}$ can be arbitrarily large, the distribution of $E_{1}\cap
E_{2}$ cannot even be supported on a compact subset of $A(n,\gamma)$. In fact,
it is known from [22, Thm. 7.2.8] that the distribution of $E_{1}\cap E_{2}$
has a non-trivial density with respect to the invariant measure on
$A(n,\gamma)$, which is proportional to

$E\mapsto\int_{A(E,q)\cap[B^{n}]_{q}}\int_{A(E,n-q+\gamma)\cap[B^{n}]_{n-q+\gamma}}[E_{1},E_{2}]^{\gamma+1}\,\mu_{n-q+\gamma}^{E}(dE_{2})\mu_{q}^{E}(dE_{1}),$

where $A(E,q)$ is the set of $q$-dimensional affine subspaces containing $E$
and $\mu_{q}^{E}$ is the invariant measure on that space (similarly for
$A(E,n-q+\gamma)$ and $\mu_{n-q+\gamma}^{E}$). Moreover, $[E_{1},E_{2}]$
stands for the so-called subspace determinant, describing the relative
position of $E_{1}$ and $E_{2}$, see below for a detailed definition. We
remark that in the special case $n=2$, $q=1$ and $\gamma=0$ this is a
classical result of M. Crofton discussed in [3, §7], whereas the case $n=3$,
$q=2$ and $\gamma=1$ goes back to W. Blaschke [3, §33].

On a more abstract level, both problems just mentioned naturally lead to the
study of what is known in the literature as integral-geometric transformation
formulas of Blaschke–Petkantschin type. Such formulas go back to the
pioneering works of W. Blaschke [3] in dimensions $n=2$, and $n=3$ and have
more systematically been investigated by his student B. Petkantschin [18].
They have further been developed in [21] using the language of differential
forms and in [22] by means of a measure-theoretic approach.
Blaschke–Petkantschin formulas are fundamental devices in integral and
stochastic geometry and have found various applications in convex geometry and
geometric analysis [4, 5, 6, 11, 16, 17, 20] as well as in stereology [7, 8,
9, 13, 14, 15]. For the intersection of two linear subspaces a formula of this
type is given by

$\begin{split}&\int_{G(n,q)}\int_{G(n,n-q+\gamma)}f(L_{1},L_{2})\,\nu_{n-q+\gamma}(dL_{2})\nu_{q}(dL_{1})\\\
&=c_{1}(n,q,\gamma)\int_{G(n,\gamma)}\int_{G(L,q)}\int_{G(L,n-q+\gamma)}f(L_{1},L_{2})[L_{1},L_{2}]^{\gamma}\,\nu_{n-q+\gamma}^{L}(dL_{2})\nu_{q}^{L}(dL_{1})\nu_{\gamma}(dL),\end{split}$
(1.1)

where $f:G(n,q)\times G(n,n-q+\gamma)\to[0,\infty)$ is a measurable function
and $G(L,q)$ is the relative Grassmannian of all $q$-dimensional linear
subspaces of $\mathbb{R}^{n}$ containing $L$, whereas $\nu_{q}^{L}$ is the
invariant probability measure on $G(L,q)$ which is invariant under all
rotations of $\mathbb{R}^{n}$ that fix $L$ (similarly for $G(L,n-q+\gamma)$
and $\nu_{n-q+\gamma}^{L}$). We refer to [22, Thm. 7.2.5] where also the value
of the constant $c_{1}(n,q,\gamma)$ can be found, which only depends on the
parameters in brackets. The corresponding formula for the intersection of two
affine subspaces is a special case of [22, Thm. 7.2.8] and reads as follows:

$\begin{split}&\int_{A(n,q)}\int_{A(n,n-q+\gamma)}f(E_{1},E_{2})\,\mu_{n-q+\gamma}(dE_{2})\mu_{q}(dE_{1})\\\
&=c_{2}(n,q,\gamma)\int_{A(n,\gamma)}\int_{A(E,q)}\int_{A(E,n-q+\gamma)}f(E_{1},E_{2})[E_{1},E_{2}]^{\gamma+1}\,\mu_{n-q+\gamma}^{E}(dE_{2})\mu_{q}^{E}(dE_{1})\mu_{\gamma}(dE)\end{split}$
(1.2)

for measurable functions $f:A(n,q)\times A(n,n-q+\gamma)\to[0,\infty)$. Here,
$A(E,k)$ stands for the family of $k$-dimensional affine subspaces containing
$E$ and $\mu_{k}^{E}$ denotes the invariant measure on $A(E,k)$, $k=q$ and
$k=n-q+\gamma$. The value of the constant $c_{2}(n,q,\gamma)$ only depends on
the parameters in brackets and can be found in [22].

The present paper deals with a situation which in a sense is intermediate
between (1.1) and (1.2), and combines the linear with the affine set-up. To
the best of our knowledge, this has not found attention so far in the
literature. More explicitly, let $L\in G(n,q)$ be a $q$-dimensional linear
subspace of $\mathbb{R}^{n}$, $q\in\\{1,\ldots,n-1\\}$, and for
$\gamma\in\\{0,\ldots,q-1\\}$ let $E\in G(n,n-q+\gamma)$ be an affine subspace
of dimension $n-q+\gamma$. If $E$ and $L$ are in general position, their
intersection $E\cap L$ is an affine subspace of dimension $\gamma$.

Our principal goal is the following. Find, for a given rotation invariant
measure $\tilde{\mu}_{n-q+\gamma}$ on $A(n,n-q+\gamma)$ that is absolutely
continuous with respect to $\mu_{n-q+\gamma}$, a measurable function
$J:A(n,\gamma)\rightarrow[0,\infty)$ such that

$\int_{G(n,q)}\int_{A(n,n-q+\gamma)}f(E\cap
L)\,\tilde{\mu}_{n-q+\gamma}(dE)\nu_{q}(dL)=\int_{A(n,\gamma)}f(E)\,J(E)\,\mu_{\gamma}(dE)$
(1.3)

holds for all measurable functions $f:A(n,\gamma)\to[0,\infty)$. The main
result of this paper provides an explicit description of $J(E)$ and its
dependence on $\tilde{\mu}_{n-q+\gamma}$. In the important particular case
$\tilde{\mu}_{n-q+\gamma}=\mu_{n-q+\gamma}$, it turns out that
$J(E)=cd(o,E)^{-(n-q)}$, where $c$ is a known constant and $d(o,E)$ stands for
the distance of $E$ to the origin $o$. Another interesting case arises when
$\tilde{\mu}_{n-q+\gamma}$ is the restriction of $\mu_{n-q+\gamma}$ to
$[hB^{n}]_{n-q+\gamma}$ for some fixed $h>0$. Then, the left-hand side of
(1.3) is

$\int_{G(n,q)}\int_{A(n,n-q+\gamma)}f(E\cap L)\,1_{\\{d(o,E)\leq h\\}}\
\mu_{n-q+\gamma}(dE)\nu_{q}(dL),$ (1.4)

and $J(E)$ involves, besides of $d(o,E)^{-(n-q)}$, an additional factor that
can be expressed in terms of an incomplete beta function. In probabilistic
terms, our new integral-geometric transformation formula (1.3) will allow us
to determine the density with respect to the invariant measure $\mu_{\gamma}$
on $A(n,\gamma)$ of the intersection of a random linear subspace $L$ of
dimension $q$ and a stochastically independent random subspace $E$ of
dimension $n-q+\gamma$ hitting the unit ball in $\mathbb{R}^{n}$, see Figure
1.1. Moreover, we will also be able to determine the corresponding density
with respect to $\mu_{\gamma}$ if the random affine subspace $E$ is only
tangent to the unit sphere, see again Figure 1.1.

The remaining parts of this paper are structured as follows. In Section 2 we
set up the notation and gather some background material. Some preliminary
considerations are contained in Section 3 and in Section 4 we formulate our
main theorems, especially the new integral-geometric transformation formula of
Blaschke–Petkantschin-type. In Sections 5 and 6 we present the two
applications to intersection probabilities mentioned above. Finally, Section 7
contains the proofs of our main results.

##  2 Notation and background material

Let $\mathbb{R}^{n}$ denote the $n$-dimensional Euclidean space for some fixed
dimension $n\geq 1$. The _Euclidean norm_ will always be denoted by
$\|\,\cdot\,\|$ and by $\lambda_{n}$ we indicate the _Lebesgue measure_ on
$\mathbb{R}^{n}$. The _Euclidean unit ball_ and _sphere_ are denoted by
$B^{n}$ and $S^{n-1}$ and their volume and surface content are given by

$\displaystyle\kappa_{n}=\lambda_{n}(B^{n})=\frac{\pi^{n/2}}{\Gamma(\frac{n}{2}+1)}\qquad\text{and}\qquad\omega_{n}=\mathcal{H}^{n-1}(S^{n-1})=\frac{2\pi^{n/2}}{\Gamma(\frac{n}{2})},$
(2.1)

respectively. Here, $\mathcal{H}^{n-1}$ is the _$(n-1)$ -dimensional Hausdorff
measure_ in $\mathbb{R}^{n}$. Further, for a set $E\subset\mathbb{R}^{n}$ we
define the distance $d(o,E)=\inf_{x\in E}\|x\|$ of $E$ to the origin
$o\in\mathbb{R}^{n}$. We will make use of the _incomplete beta function_

$B(x;\alpha,\beta)=\int_{0}^{x}t^{\alpha-1}(1-t)^{\beta-1}\,dt,\qquad 0\leq
x\leq 1,$

with real parameters $\alpha,\beta>0$. Notice that the _complete beta
integral_ satisfies

$B\big{(}\tfrac{m}{2},\tfrac{k}{2}\big{)}=B\big{(}1;\tfrac{m}{2},\tfrac{k}{2}\big{)}=2\frac{\omega_{m+k}}{\omega_{m}\omega_{k}},\qquad
m,k\in\mathbb{N}.$ (2.2)

For $n\geq 1$ and $q\in\\{0,\ldots,n\\}$ we let $G(n,q)$ be the _Grassmannian_
of all $q$-dimensional linear subspaces of $\mathbb{R}^{n}$ and write $A(n,q)$
for the family of $q$-dimensional affine subspaces of $\mathbb{R}^{n}$. We
endow these spaces with the usual Borel $\sigma$-algebras and describe now
shortly invariant measures on these families. Details can be found e.g. in
[22], where also measurablility issues are discussed.

The group $SO_{n}$ of rotations in $\mathbb{R}^{n}$ carries a unique invariant
(or _Haar_) probability measure $\nu$. This group acts naturally on the space
$G(n,q)$ and we denote by $\nu_{q}$ the unique $SO_{n}$-invariant probability
measure on $G(n,q)$. Both, $SO_{n}$ and the group of translations act
naturally on the _affine Grassmannian_ $A(n,q)$. There exists a motion
invariant measure on $A(n,q)$, and this measure is unique up to a
multiplicative constant. We will use the motion invariant measure $\mu_{q}$ on
the affine Grassmannian $A(n,q)$ given by

$\displaystyle\mu_{q}(\,\cdot\,)=\int_{G(n,q)}\int_{L^{\perp}}1_{\\{L+x\in\,\cdot\,\\}}\
\lambda_{L^{\perp}}(dx)\nu_{q}(dL),$ (2.3)

where $\lambda_{L^{\perp}}$ denotes the Lebesgue measure on the orthogonal
complement $L^{\perp}$ of $L\in G(n,q)$. Finally, for $M\in A(n,q)$ and
$p\in\\{0,\ldots,q\\}$ we denote by $G(M,p)$ and $A(M,p)$ the _relative
Grassmannian_ of all $p$-dimensional linear and affine subspaces contained in
$M$, respectively. If, on the other hand, $p\in\\{q,\ldots,n\\}$ then $G(M,p)$
and $A(M,p)$ are the sets of linear and affine subspaces of dimension $p$ that
contain $M$. These spaces carry natural invariant measures $\nu_{p}^{M}$ and
$\mu_{p}^{M}$ as described in [22, Sec. 7.1]. In particular, these measures
satisfy

$\displaystyle\int_{G(n,q)}\int_{G(M,p)}f(L)\ \nu_{p}^{M}(dL)\
\nu_{q}(dM)=\int_{G(n,p)}f(L)\ \nu_{p}(dL)$

for all measurable functions $f:G(n,p)\to[0,\infty)$, and similarly

$\displaystyle\int_{A(n,q)}\int_{A(F,p)}f(E)\ \mu_{p}^{F}(dE)\
\mu_{q}(dF)=\int_{A(n,p)}f(E)\ \mu_{p}(dE)$

for all measurable functions $f:A(n,p)\to[0,\infty)$, see [22, Thm. 7.1.1 and
Thm. 7.1.2].

Let $0\leq p,q\leq n-1$ and fix $L\in G(n,p)$ and $M\in G(n,q)$. If $p+q\leq
n$ the _subspace determinant_ $[L,M]$ is defined as the $(p+q)$-volume of a
parallelepiped spanned by the union of an orthonormal basis in $L$ and an
orthonormal basis in $M$. If $p+q\geq n$ we define
$[L,M]=[L^{\perp},M^{\perp}]$. If $p+q=n$, both definitions coincide and
$[L,M]$ is the factor by which the $p$-volume is multiplied under the
orthogonal projection from $L$ onto $M^{\perp}$ thus,

$\int_{L^{\perp}}f(x)\ \lambda_{L^{\perp}}(dx)=[M,L]\int_{M}f(x|L^{\perp})\
\lambda_{M}(dx),$ (2.4)

for measurable $f:L^{\perp}\to[0,\infty)$. Here, $x|L^{\perp}$ denotes the
orthogonal projection of a point $x\in\mathbb{R}^{n}$ onto $L^{\perp}$. For
further background on subspace determinants, we refer to [22, Sec. 14.1].

##  3 Preliminary considerations

Let $n\geq 1$ and $q\in\\{1,\ldots,n-1\\}$. The rotation group $SO_{n}$ acts
on the space of real-valued functions $f$ on $A(n,q)$ by

$(\eta f)(E)=f(\eta^{-1}E),\qquad E\in A(n,q),$

for $\eta\in SO_{n}$. The function $f$ is called _rotation invariant_ if $\eta
f=f$ for any rotation $\eta\in SO_{n}$. The _rotational mean_ of $f$, given by

$f_{\mbox{\small rot}}(E)=\int_{SO_{n}}(\eta f)(E)\ \nu(d\eta),\qquad E\in
A(n,q),$

is rotation invariant.

Assume that $f$ is rotation invariant and $n\geq 2$. Then, we have
$f(E)=f(E^{\prime})$ for any two affine subspaces $E,E^{\prime}\in A(n,q)$
with $d(o,E)=d(o,E^{\prime})$, since there is a rotation $\eta\in SO_{n}$ with
$\eta E=E^{\prime}$. This implies that there is a function
$f_{I}:[0,\infty)\to\mathbb{R}$, such that

$f(E)=f_{I}(d(o,E)),\qquad E\in A(n,q).$ (3.1)

The following lemma shows that for our purposes, it is essentially enough to
consider the class of rotation invariant functions.

######  Lemma 3.1.

Let $n\geq 2$, $q\in\\{1,\ldots,n-1\\}$, $\gamma\in\\{0,\ldots,q-1\\}$, a
rotation invariant measure $\tilde{\mu}_{n-q+\gamma}$ on $A(n,n-q+\gamma)$ and
a rotation invariant function $J:A(n,\gamma)\rightarrow[0,\infty)$ be given.
Then

$\int_{G(n,q)}\int_{A(n,n-q+\gamma)}f(E\cap L)\ \tilde{\mu}_{n-q+\gamma}(dE)\
\nu(dL)=\int_{A(n,\gamma)}f(E)\ J(E)\ \mu_{\gamma}(dE)$ (3.2)

holds for all measurable functions $f:A(n,\gamma)\rightarrow[0,\infty)$, if it
holds for those $f$ that are in addition rotation invariant.

###### Proof.

Suppose that (3.2) holds true for all non-negative, measurable and rotation
invariant functions. Fix an arbitrary measurable function
$f:A(n,q)\rightarrow[0,\infty)$. Using the rotation invariance of the measures
$\tilde{\mu}_{n-q+\gamma}$ and $\nu_{q}$ and Tonelli’s theorem, the left-hand
side of (3.2) is

$\displaystyle\int_{SO_{n}}\int_{G(n,q)}\int_{A(n,n-q+\gamma)}f(\eta^{-1}(E\cap
L))\ \tilde{\mu}_{n-q+\gamma}(dE)\ \nu_{q}(dL)\nu(d\eta)$
$\displaystyle=\int_{G(n,q)}\int_{A(n,n-q+\gamma)}f_{\mbox{\small rot}}(E)\
\tilde{\mu}_{n-q+\gamma}(dE)\ \nu_{q}(dL).$

Using the rotation invariance of $\mu_{\gamma}$ and $J$, a similar argument
shows that the right-hand side of (3.2) is

$\displaystyle\int_{A(n,\gamma)}f_{\mbox{\small
rot}}(E)J(E)\,\mu_{\gamma}(dE).$

By assumption, (3.2) holds for the rotation invariant function
$f_{\mbox{\small rot}}$, so the last two displayed expressions coincide and
the assertion is shown. ∎

In the proof of our main result, the following lemma will turn out to be of
crucial importance. It can be seen a generalization of [12, Lem. 4.4]. In that
result the authors prove that

$A(n,k,r,\alpha)=\int_{G(n,k)}[F,L]^{\alpha}\,\nu_{k}(dL)=\prod_{i=0}^{n-r-1}\frac{\Gamma(\frac{n-i}{2})\Gamma(\frac{k-i+\alpha}{2})}{\Gamma(\frac{n-i+\alpha}{2})\Gamma(\frac{k-i}{2})},$
(3.3)

for $\alpha\geq 0$, $r,k\in\\{1,\ldots,n\\}$ with $r+k\geq n$ and $F\in
G(n,r)$, where the right-hand side is interpreted as $1$ if $r=n$. The next
lemma is a counterpart for the Grassmannian associated to a hyperplane. (We
note that the definition of the subspace determinant in [12] is equivalent to
our definition as $r+k\geq n$.)

######  Lemma 3.2.

Let $n\geq 2$, $\alpha\geq 0$ and $p,q\in\\{1,\ldots,n-1\\}$ with $p+q\leq n$
be given. Then, for any $u\in S^{n-1}$ and any fixed $M\in G(n,p)$ we have
that

$\int_{G(\operatorname{span}u,q)}[L,M]^{\alpha}\,\nu_{q}^{\operatorname{span}{u}}(dL)=a({n,p,q,\alpha})[u,M]^{\alpha}$
(3.4)

with

$a({n,p,q,\alpha})=\prod_{i=1}^{p}\frac{\Gamma(\frac{n-i}{2})\Gamma(\frac{n-q-i+\alpha+1}{2})}{\Gamma(\frac{n-i+\alpha}{2})\Gamma(\frac{n-q-i+1}{2})}.$

###### Proof.

If $u\in M$, then (3.4) holds trivially as both sides vanish. Hence, we may
assume $u\not\in M$. Suppose first that we also have $u\not\in M^{\perp}$ and
fix $L\in G(\operatorname{span}{u},q)$. Since $\dim M^{\perp}+\dim
L^{\perp}=2n-(p+q)\geq n$, [19, Lem. 4.1] implies that

$\displaystyle[L^{\perp},M^{\perp}]$ $\displaystyle=[L^{\perp},M^{\perp}\cap
u^{\perp}]\|u|M^{\perp}\|$ $\displaystyle=[L^{\perp},M^{\perp}\cap
u^{\perp}][u,M],$

where we recall that $u|M^{\perp}$ is the projection of $u$ onto $M^{\perp}$.
Here, we used that fact that the subspace determinant
$[L^{\prime},M^{\prime}]$, as defined in [19], coincides with our definition
whenever $\dim M^{\prime}+\dim L^{\prime}\geq n$, but differs otherwise,
causing [19] to consider subspace determinants relative to $u^{\perp}$, which
is not necessary using our definition. We thus obtain

$[L,M]=[L^{\perp},M^{\perp}]=[L^{\perp},M^{\perp}\cap u^{\perp}][u,M],$

a relation that is also true for $u\in M^{\perp}$, implying that the left-hand
side of (3.4) coincides with

$\displaystyle\int_{G(\operatorname{span}{u},q)}[L^{\perp},M^{\perp}\cap
u^{\perp}]^{a}\,\nu_{q}^{\operatorname{span}{u}}(dL)\,[u,M]^{\alpha}=\int_{G(u^{\perp},n-q)}[L,M^{\perp}\cap
u^{\perp}]^{a}\,\nu_{q-1}^{u^{\perp}}(dL)\,[u,M]^{\alpha}.$

The last integral is now of the form (3.3), but with $u^{\perp}$ instead of
$\mathbb{R}^{n}$ as the ambient space. It is thus equal to
$A(n-1,n-q,n-p-1,\alpha)$. This constant coincides with $a({n,p,q,\alpha})$,
and the assertion is proven. ∎

##  4 Presentation of the main results

Having established the basic notions and concepts in Section 3, we will now
state our main result: a general reduction of integrals of the form (1.3). We
remark that we imposed the dimensional constraints $0\leq\gamma<q<n$ at the
beginning of the introduction, since $\gamma=q$ or $q=n$, would imply that the
left-hand side of (1.3) becomes trivial. These constraints imply the
assumption $n\geq 2$, which we now adopt for the rest of the paper.

Recall that (1.3) involves a rotation invariant measure
$\tilde{\mu}_{n-q+\gamma}$ on $A(n,n-q+\gamma)$ that is dominated by
$\mu_{n-q+\gamma}$. Hence, the Radon–Nikodym theorem guarantees the existence
of a $\mu_{n-q+\gamma}$-density $\tilde{H}\geq 0$ for
$\tilde{\mu}_{n-q+\gamma}$. The assumed rotation invariance implies that
$H=\tilde{H}_{\mbox{\small rot}}$ is also a $\mu_{n-q+\gamma}$-density for
$\tilde{\mu}_{n-q+\gamma}$. We will state our results using this density, as
the function $J(r)=J_{H}(r)$ can explicitly be expressed in terms of $H$. For
specific choices of the density $H$, the function $J_{H}(r)$ in the statement
of this theorem can be simplified and be made more explicit, as illustrated in
Corollaries 4.2 and 4.4 below. The proof of Theorem 4.1 is postponed to
Section 7 at the end of this paper.

######  Theorem 4.1.

Fix $n\geq 2$, $q\in\\{1,\ldots,n-1\\}$, $\gamma\in\\{0,\ldots,q-1\\}$, and
let $H:A(n,n-q+\gamma)\rightarrow[0,\infty)$ be a measurable and rotation
invariant function. Then

$\displaystyle\int_{G(n,q)}\int_{A(n,n-q+\gamma)}f(E\cap L)H(E)\
\mu_{n-q+\gamma}(dE)\ \nu_{q}(dL)$
$\displaystyle=D(n,q,\gamma)\int_{A(n,\gamma)}f(E)d(o,E)^{-(n-q)}J_{H}(d(o,E))\
\mu_{\gamma}(dE)$

for all measurable $f:A(n,\gamma)\rightarrow[0,\infty)$. Here,

$J_{H}(r)=\int_{0}^{1}H_{I}(rz)z^{q}(1-z^{2})^{\frac{n-q}{2}-1}\ dz,$ (4.1)

and the constant is given by

$D(n,q,\gamma)=\frac{\omega_{\gamma+1}\omega_{q-\gamma}\omega_{n-q}}{\omega_{n-(q-\gamma)+1}\omega_{n-\gamma}}.$
(4.2)

We shall now discuss two special cases in which the function $J_{H}(r)$ and
thus the integral relation in Theorem 4.1 can be simplified. We start by
considering the special case, where $H$ is the function
$H_{h}(E)=1_{\\{d(o,E)\leq h\\}}$ for some $h>0$, corresponding to
$\tilde{\mu}_{n-q+\gamma}$ in (1.3) being the restriction of
$\mu_{n-q+\gamma}$ to $[hB^{n}]_{n-q+\gamma}$. Definition (4.1) and a
substitution yield

$\displaystyle J_{H_{h}}(r)$
$\displaystyle=\frac{1}{2}\int_{0}^{(\min\\{\frac{h}{r},1\\})^{2}}z^{\frac{q+1}{2}-1}(1-z)^{\frac{n-q}{2}-1}\
dz=\begin{cases}\frac{\omega_{n+1}}{\omega_{q+1}\omega_{n-q}}&:r\leq h,\\\
\frac{1}{2}B\big{(}(\tfrac{h}{r})^{2};\frac{q+1}{2},\frac{n-q}{2}\big{)}&:r>h,\end{cases}$
(4.3)

where (2.2) was used at the second equality sign. This implies the following
result.

######  Corollary 4.2.

Fix $n\geq 2$, $q\in\\{1,\ldots,n-1\\}$, $\gamma\in\\{0,\ldots,q-1\\}$, and
$h>0$. Then

$\displaystyle\int_{G(n,q)}\int_{[hB^{n}]_{n-q+\gamma}}f(E\cap
L)\,\mu_{n-q+\gamma}(dE)\,\nu_{q}(dL)$
$\displaystyle=D(n,q,\gamma)\int_{A(n,\gamma)}f(E)d(o,E)^{-(n-q)}J(d(o,E))\,\mu_{\gamma}(dE),$

where $J=J_{H_{h}}$ is given by (4.3) and where $D(n,q,\gamma)$ is the
constant given in Theorem 4.1.

The incomplete beta integral, and hence the function $J$ in Corollary 4.2, can
be expressed in terms of a hypergeometric function. Also, weight functions
$J_{H}$ associated to more general densities $H$ can be expressed in terms of
– possibly several – hypergeometric functions. For instance, if $H_{I}$ is an
even polynomial restricted to $[0,\infty)$, Euler’s integral relation implies
such a representation, see e.g. [1] for details.

Using the monotone convergence theorem when $h\to\infty$ in Corollary 4.2,
gives an explicit integral relation of the form (1.3) with
$\tilde{\mu}_{n-q+\gamma}=\mu_{n-q+\gamma}$.

######  Corollary 4.3.

Fix $n\geq 2$, $q\in\\{1,\ldots,n-1\\}$ and $\gamma\in\\{0,\ldots,q-1\\}$.
Then

$\displaystyle\int_{G(n,q)}\int_{A(n,n-q+\gamma)}f(E\cap
L)\,\mu_{n-q+\gamma}(dE)\,\nu_{q}(dL)$
$\displaystyle=\tilde{D}(n,q,\gamma)\int_{A(n,\gamma)}f(E)d(o,E)^{-(n-q)}\
\mu_{\gamma}(dE)$

for all measurable functions $f:A(n,\gamma)\rightarrow[0,\infty)$. Here,

$\tilde{D}(n,q,\gamma)=\frac{\omega_{n+1}\omega_{\gamma+1}\omega_{q-\gamma}}{\omega_{n-(q-\gamma)+1}\omega_{n-\gamma}\omega_{q+1}}.$

Let us also report that Theorem 4.1 allows an extension to multiple
intersections in the following way. The proof is postponed to Section 7.

######  Corollary 4.4.

Fix $\ell,m,n\geq 1$, $q_{1},\ldots,q_{\ell}\in\\{0,\ldots,n\\}$, put
$q=q_{1}+\dots+q_{\ell}-(\ell-1)n$. Suppose that $q\in\\{1,\ldots,n-1\\}$ and
fix $\gamma\in\\{0,\ldots,q-1\\}$ and $p_{1},\ldots,p_{m}\in\\{0,\ldots,n\\}$
such that $p_{1}+\dots+p_{m}-(m-1)n=n-q+\gamma$. Further, let
$f:A(n,\gamma)\to[0,\infty)$ be a measurable function, and let
$H:A(n,n-q+\gamma)\to[0,\infty)$ be a rotation invariant measurable function.
Define

$\displaystyle I_{\ell,m}$
$\displaystyle=\int_{G(n,q_{1})}\cdots\int_{G(n,q_{\ell})}\int_{{A(n,p_{1})}}\cdots\int_{A(n,p_{m})}f(E_{1}\cap\ldots\cap
E_{m}\cap L_{1}\cap\ldots\cap L_{\ell})$ $\displaystyle\hskip 85.35826pt\times
H(E_{1}\cap\ldots\cap
E_{m})\,\mu_{p_{m}}(dE_{m})\dots\mu_{p_{1}}(dE_{1})\nu_{q_{\ell}}(dL_{\ell})\dots\nu_{q_{1}}(dL_{1}).$

Then

$I_{\ell,m}=D(n,q,\gamma){\omega_{n-q+\gamma+1}\omega_{n+1}^{m}\over\omega_{n+1}\omega_{p_{1}}\cdots\omega_{p_{m}}}\,\int_{A(n,\gamma)}f(E)d(o,E)^{-(n-q)}J_{H}(d(o,E))\
\mu_{\gamma}(dE),$

where $J_{H}$ is given by (4.1) and the leading constant by (4.2).

Finally, we state an alternative version of Theorem 4.1, which appears to be
more general, as the intersecting linear subspace now is fixed. Closer
investigation shows, however, that the two statements are equivalent, if a
suitable Blaschke–Petkantschin relation is applied. The proof of this
equivalence will be given in Section 7.

######  Theorem 4.5.

Fix $n\geq 2$, $q\in\\{1,\ldots,n-1\\}$, $\gamma\in\\{0,\ldots,q-1\\}$ and
$L_{0}\in G(n,q)$. Let $H:A(n,n-q+\gamma)\rightarrow[0,\infty)$ be a rotation
invariant, measurable function. Then

$\displaystyle\int_{A(n,n-q+\gamma)}f(E\cap L_{0})H(E)\
\mu_{n-q+\gamma}(dE)=\frac{\omega_{\gamma+1}\omega_{n-q}}{\omega_{n-(q-\gamma)+1}}\int_{A(L_{0},\gamma)}f(E)J_{H}(d(o,E))\
\mu_{\gamma}^{L_{0}}(dE)$ (4.4)

for all measurable $f:A(L_{0},\gamma)\rightarrow[0,\infty)$. Here, $J_{H}$ is
given in Theorem 4.1.

Similar versions, that is, versions where the linear space is hold fixed, are
possible also for Corollaries 4.2-4.4.

##  5 Intersection probabilities for linear and affine subspaces hitting the
unit ball

In this section we return to the problem from stochastic geometry already
mentioned in the introduction. Namely, we consider a random linear subspace
$L$ of dimension $q\in\\{1,\ldots,n-1\\}$ in $\mathbb{R}^{n}$ with
distribution $\nu_{q}$ and a random affine subspace $E$ of dimension
$n-q+\gamma$, where $\gamma\in\\{0,\ldots,q-1\\}$ is a fixed number. Recalling
that

$[B^{n}]_{n-q+\gamma}=\\{E\in A(n,n-q+\gamma):E\cap B^{n}\neq\emptyset\\},$

we use the restriction of $\kappa_{q-\gamma}^{-1}\mu_{n-q+\gamma}$ to
$[B^{n}]_{n-q+\gamma}$ as distribution for $E$ and suppose that $L$ and $E$
are stochastically independent. In fact, it follows from (2.3) that
$\kappa_{q-\gamma}^{-1}\mu_{n-q+\gamma}$ is indeed a probability measure on
$[B^{n}]_{n-q+\gamma}$. We are interested in the distribution of the random
affine subspace $E\cap L$, which is almost surely of dimension $\gamma$. Since
its distribution is invariant under rotations of $\mathbb{R}^{n}$, all
relevant information is contained in the distribution of the random variable
$d(o,E\cap L)$ describing the distance of $E\cap L$ to the origin. This
distribution turns out to have a density $f_{n,q,\gamma}$ with respect to
Lebesgue measure on $[0,\infty)$. The next result shows that it is heavy
tailed and asymptotically of Pareto type with shape parameter $\gamma+2$.

######  Theorem 5.1.

In the set-up just introduced, the probability density
$f_{n,q,\gamma}(\delta)$ of the random variable $d(o,E\cap L)$ is given by

$\displaystyle
f_{n,q,\gamma}(\delta)=(q-\gamma){\omega_{\gamma+1}\omega_{n-q}\over\omega_{n-(q-\gamma)+1}}\begin{cases}{\omega_{n+1}\over\omega_{q+1}\omega_{n-q}}\delta^{q-\gamma-1}&:0\leq\delta\leq
1\\\ {1\over 2}\delta^{q-\gamma-1}B({1\over\delta^{2}};{q+1\over 2},{n-q\over
2})&:\delta>1.\end{cases}$

###### Proof.

Fix $\delta>0$ and consider the probability of the event $d(o,E\cap
L)\leq\delta$. Using Corollary 4.2 with $h=1$, we obtain

$\displaystyle\mathbb{P}[d(o,E\cap L)\leq\delta]$
$\displaystyle={1\over\kappa_{q-\gamma}}\int_{G(n,q)}\int_{A(n,n-q+\gamma)}1_{\\{d(o,E)\leq
1\\}}\,1_{\\{d(o,E\cap L)\leq\delta\\}}\,\mu_{n-q+\gamma}(dE)\nu_{q}(dL)$
$\displaystyle={D(n,q,\gamma)\over\kappa_{q-\gamma}}\int_{A(n,\gamma)}1_{\\{d(o,E)\leq\delta\\}}d(o,E)^{-(n-q)}J_{H_{1}}(d(o,E))\,\mu_{\gamma}(dE),$

with

$J_{H_{1}}(r)=\begin{cases}{\omega_{n+1}\over\omega_{q+1}\omega_{n-q}}&:r\leq
1\\\ {1\over 2}B({1\over r^{2}};{q+1\over 2},{n-q\over 2})&:r>1.\end{cases}$

We now apply the decomposition (2.3), use $d(o,L+x)=\|x\|$ whenever $x\in
L^{\perp}$, and then introduce spherical coordinates in $L^{\perp}$ to see
that

$\displaystyle\mathbb{P}[d(o,E\cap L)\leq\delta]$
$\displaystyle={D(n,q,\gamma)\over\kappa_{q-\gamma}}\int_{G(n,\gamma)}\int_{L^{\perp}}1_{\\{\|x\|\leq\delta\\}}\|x\|^{-(n-q)}J_{H_{1}}(\|x\|)\,\lambda_{L^{\perp}}(dx)\nu_{\gamma}(dL)$
$\displaystyle={D(n,q,\gamma)\omega_{n-\gamma}\over\kappa_{q-\gamma}}\int_{0}^{\delta}r^{q-\gamma-1}J_{H_{1}}(r)\,dr.$

Taking the derivative with respect to $\delta$ in the last expression,
inserting the value of the constant $D(n,q,\gamma)$ of Theorem 4.1 and then
using (2.1) yields the result. ∎

Although the density $f_{n,q,\gamma}(\delta)$ is defined piecewise, we remark
that it is continuous at the point $\delta=1$. In fact, this follows from the
continuity of the incomplete beta function in combination with (2.2). Let us
also mention at this point that since we are working with the unit ball
$B^{n}$ as a reference set, the results of this section and also the next one
continue to hold if the random linear subspace $L$ is replaced by a
deterministic linear subspace of the same dimension.

Since the density of $d(o,E\cap L)$ is asymptotically of Pareto type with
shape parameter $\gamma+2$, the moment properties of the random variable
$d(o,E\cap L)$ depend on the intersection dimension $\gamma$. The next result
delivers a precise description.

######  Corollary 5.2.

Let $\alpha\in\mathbb{R}$. In the set-up just introduced, one has that
$\mathbb{E}d(o,E\cap L)^{\alpha}<\infty$ if and only if
$\alpha\in(\gamma-q,\gamma+1)$. In particular, if $\gamma=0$ the random
variable $d(o,E\cap L)$ has infinite expectation.

###### Proof.

We have to check under which conditions on $\alpha$ the product of
$\delta^{\alpha}$ with the probability density $f_{n,q,\gamma}(\delta)$ of
Theorem 5.1 is integrable at $\delta=0$ and $\delta=\infty$. The function
$\delta\mapsto\delta^{q-\gamma-1+\alpha}$ is integrable at $\delta=0$, if and
only if $\alpha>\gamma-q$. Moreover, since
$B(\delta^{-2};a,b)={\delta^{-2a}\over a}+O(\delta^{-2(a+1)})$ as
$\delta\to\infty$, see [1, §6.6.8 and §15.7], the required integrability is
satisfied, whenever the function
$\delta^{\alpha+q-\gamma-1-(q+1)}=\delta^{\alpha-\gamma-2}$ is integrable at
$\delta=\infty$. The latter holds if and only if $\alpha-\gamma-2<-1$, or
equivalently, $\alpha<\gamma+1$. This completes the proof. ∎

As anticipated above, the intersection of $E$ and $L$ may or may not hit the
unit ball. By Theorem 5.1 the probability for the first of these events is
given by

$\displaystyle\mathbb{P}[E\cap L\cap B^{n}\neq\emptyset]$
$\displaystyle=(q-\gamma){\omega_{\gamma+1}\omega_{n-q}\over\omega_{n-(q-\gamma)+1}}{\omega_{n+1}\over\omega_{q+1}\omega_{n-q}}\int_{0}^{1}\delta^{q-\gamma-1}\,d\delta.$

Simplification leads to the following result.

######  Corollary 5.3.

In the set-up just introduced, it holds that

$p_{n,q,\gamma}=\mathbb{P}[E\cap L\cap
B^{n}\neq\emptyset]={\omega_{\gamma+1}\omega_{n+1}\over\omega_{q+1}\omega_{n-(q-\gamma)+1}}.$

This result allows quantifying the asymptotic behavior of the intersection
probability $p_{n,q,\gamma}$ for fixed $q$ and $\gamma$ in high dimensions,
that is, as $n\to\infty$. Using (2.1), we can rewrite $p_{n,q,\gamma}$ in
terms of gamma functions as

$\displaystyle p_{n,q,\gamma}={\Gamma({q+1\over 2})\Gamma({n-(q-\gamma)+1\over
2})\over\Gamma({\gamma+1\over 2})\Gamma({n+1\over 2})}.$

Since $q$ and $\gamma$ are assumed to be fixed, we can apply Stirling’s
formula [1, §6.1.37] for the gamma function to see that

$p_{n,q,\gamma}={\Gamma({q+1\over 2})\over\Gamma({\gamma+1\over
2})}\Big{(}{2\over n}\Big{)}^{q-\gamma\over
2}(1+o_{n}(1))={\omega_{\gamma+1}\over\omega_{q+1}}\Big{(}{2\over\pi\,n}\Big{)}^{q-\gamma\over
2}(1+o_{n}(1)),$

as $n\to\infty$, where we write $o_{n}(1)$ for a sequence that converges to
zero with $n$. Hence, in high dimensions, the intersection point of the random
affine subspace $E$ and the random linear subspace $L$ will asymptotically
almost surely be outside the unit ball $B^{n}$.

Figure 5.1: First line: The probability densities $f_{2,1,0}$, $f_{3,2,1}$ and
$f_{3,1,0}$ from left to right. Second line: The probability densities
$f_{8,5,2}$, $f_{9,5,3}$ and $f_{9,6,1}$ from left to right.

In the planar case $n=2$ we can take $q=1$ and $\gamma=2$ and obtain

$\displaystyle f_{2,1,0}(\delta)$
$\displaystyle={2\over\pi}\,\begin{cases}1&:0\leq\delta\leq 1\\\
1-\sqrt{1-\delta^{-2}}&:\delta>1,\end{cases}\qquad\qquad$ $\displaystyle
p_{2,1,0}$ $\displaystyle={2\over\pi}.$ Moreover, if $n=3$ we can consider the
case $q=2$ and $\gamma=1$, which yields $\displaystyle f_{3,2,1}(\delta)$
$\displaystyle=\begin{cases}{\pi\over 4}&:0\leq\delta\leq 1\\\ {1\over
2}(\operatorname{arccsc}(\delta)-\sqrt{1-\delta^{-2}})&:\delta>1,\end{cases}\qquad\qquad$
$\displaystyle p_{3,2,1}$ $\displaystyle={\pi\over 4},$ and the case $q=1$ and
$\gamma=0$, where the probability density reduces to $\displaystyle
f_{3,1,0}(\delta)$ $\displaystyle={1\over 2}\begin{cases}1&:0\leq\delta\leq
1\\\ \delta^{-2}&:\delta>1,\end{cases}\qquad\qquad$ $\displaystyle p_{3,1,0}$
$\displaystyle={1\over 2},$

see also Figure 5.1. In particular, if $n=3$, $q=2$ and $\gamma=1$, we obtain

$\mathbb{E}d(o,E\cap L)=\int_{0}^{\infty}\delta
f_{3,2,1}(\delta)\,d\delta={\pi\over 4},$

whereas in the other two cases $\mathbb{E}d(o,E\cap L)=\infty$. To illustrate
the potential complexity of the density functions $f_{n,q,\gamma}(\delta)$ we
also record the following values:

$n,q,\gamma$ | $f_{n,q,\gamma}$ | $p_{n,q,\gamma}$ | $\mathbb{E}d(o,E\cap L)$
---|---|---|---
$8,5,2$ | $\begin{cases}{128\over 35\pi}\delta^{2}&:0\leq\delta\leq 1\\\ {48\over 105\pi}{8\delta^{7}-(8\delta^{6}+4\delta^{4}+3\delta^{2}-15)\sqrt{\delta^{2}-1}\over\delta^{5}}&:\delta>1,\end{cases}$ | ${128\over 105\pi}$ | ${4\over\pi}$
$9,5,3$ | $\begin{cases}\delta&:0\leq\delta\leq 1\\\ {4\delta^{2}-3\over\delta^{7}}&:\delta>1,\end{cases}$ | ${1\over 2}$ | ${16\over 15}$
$9,6,1$ | $\begin{cases}{75\pi\over 265}\delta^{4}&:0\leq\delta\leq 1\\\ {75\delta^{8}\operatorname{arccsc}(\delta)-5(16\delta^{6}+10\delta^{4}+8\delta^{2}-48)\sqrt{\delta^{2}-1}\over 128\delta^{4}}&:\delta>1.\end{cases}$ | ${15\pi\over 256}$ | ${5\pi\over 8}$

We see that the complexity of the probability density $f_{n,q,\gamma}$ varies
depending on $n,q,\gamma$. We notice in particular that
$p_{9,5,3}=\frac{1}{2}$, which is the same as $p_{3,1,0}$ from before. In
fact, for all odd $n$, there are (often multiple) ways we can choose $q$ and
$\gamma$ such that $p_{n,q,\gamma}=\frac{1}{2}$. For instance, if we take
$\gamma=\frac{n-1}{2}-1$ and $q=\frac{n-1}{2}+1$ or $q=n-2$, then
$p_{n,q,\gamma}=\frac{1}{2}$.

##  6 Intersection probabilities for linear and affine subspaces tangent to
the unit sphere

In this section, we consider another application of Theorem 4.1 to
intersection probabilities in stochastic geometry. As in the previous section,
we consider a random linear subspace $L$ of dimension $q\in\\{1,\ldots,n-1\\}$
in $\mathbb{R}^{n}$ with distribution $\nu_{q}$. In contrast, $E$ is now a
stochastically independent random affine subspace $E$ in $\mathbb{R}^{n}$ of
dimension $n-q+\gamma$ such that $d(o,E)=1$. Thus, $E$ is tangent to the unit
sphere $S^{n-1}$ in $\mathbb{R}^{n}$. The rotation invariant probability
measure on that space is given by

$\sigma_{n-q+\gamma}(A)=\frac{1}{\omega_{q-\gamma}}\int_{G(n,n-q+\gamma)}\int_{S^{n-1}\cap
M^{\perp}}1_{A}(M+u)\ \mathcal{H}^{q-\gamma-1}(du)\ \nu_{n-q+\gamma}(dM)$
(6.1)

for a Borel sets $A\subseteq\\{E^{\prime}\in A(n,n-q+\gamma)\ :\
d(o,E^{\prime})=1\\}$. As in the previous section, we are interested in the
distribution of $L\cap E$. Again, all relevant information is contained in the
distribution of $d(o,E\cap L)$ due to rotation invariance. The next result is
a stepping stone in the derivation of the probability density of the random
variable $d(o,E\cap L)$.

######  Lemma 6.1.

Fix $n\geq 2$, $q\in\\{1,\ldots,n-1\\}$, $\gamma\in\\{0,\ldots,q-1\\}$ and let
$f:A(n,\gamma)\rightarrow[0,\infty)$ be a measurable, rotation invariant and
bounded function. Then

$\displaystyle\int_{G(n,q)}$
$\displaystyle\int_{G(n,n-q+\gamma)}\int_{S^{n-1}\cap M^{\perp}}f((M+hu)\cap
L)\ \mathcal{H}^{q-\gamma-1}(du)\ \nu_{n-q+\gamma}(dM)\ \nu_{q}(dL)$
$\displaystyle=D(n,q,\gamma){\omega_{n-\gamma}}h^{\gamma+1}\int_{h}^{\infty}f_{I}(r)r^{-(\gamma+2)}\Big{(}1-\frac{h^{2}}{r^{2}}\Big{)}^{\frac{n-q}{2}-1}\
dr$

for almost every $h>0$.

###### Proof.

Let $f:A(n,\gamma)\rightarrow[0,\infty)$ satisfy the above assumptions and
define the function $F:(0,\infty)\to[0,\infty)$ by

$\displaystyle F(h)$
$\displaystyle=D(n,q,\gamma)\int_{A(n,\gamma)}f(E)d(o,E)^{-(n-q)}J_{H_{h}}(d(o,E))\
\mu_{\gamma}(dE)$
$\displaystyle=D(n,q,\gamma)\omega_{n-\gamma}\int_{0}^{\infty}f_{I}(r)r^{q-\gamma-1}J_{H_{h}}(r)\
dr,$

with $D(n,q,\gamma)$ as in Theorem 4.1 and $J_{H_{h}}$ given by (4.3). Since

$\displaystyle\frac{\partial}{\partial h}J_{H_{h}}(r)=\begin{cases}0&:0\leq
r<h\\\
r^{-q-1}h^{q}\Big{(}1-\frac{h^{2}}{r^{2}}\Big{)}^{\frac{n-q}{2}-1}&:r>h,\end{cases}$

we conclude that $F(h)$ is differentiable for almost all $h>0$ with derivative

$F^{\prime}(h)=D(n,q,\gamma)\omega_{n-\gamma}\int_{h}^{\infty}f_{I}(r)r^{-\gamma-2}h^{q}\Big{(}1-\frac{h^{2}}{r^{2}}\Big{)}^{\frac{n-q}{2}-1}\
dr.$

Here, differentiation under the integral sign can be justified as follows. For
fixed $r>0$, the integrand $g(r,h)=f_{I}(r)r^{q-\gamma-1}J_{H_{h}}(r)$ is
absolutely continuous as function of $h$. Since
$(r,h)\mapsto\partial/(\partial h)g(r,h)$ is integrable on
$(0,\infty)\times(a,b)$ for any $0<a<b<\infty$, Fubini’s theorem and Lebesgue
differentiation theorem imply

$F^{\prime}(h)=\frac{d}{dh}\int_{1}^{h}\int_{0}^{\infty}\frac{\partial}{\partial
h}g(r,t)drdt=\int_{0}^{\infty}\frac{\partial}{\partial h}g(r,h)dr$

for a.e. $h\in(a,b)$, yielding the intermediate assertion.

On the other hand, Theorem 4.1, relation (2.3) and spherical integration yield

$F(h)=\int_{G(n,q)}\int_{G(n,n-q+\gamma)}\int_{S^{n-1}\cap
M^{\perp}}\int_{0}^{h}f((M+ru)\cap L)r^{q-\gamma-1}\ dr\
\mathcal{H}^{q-\gamma-1}(du)\ \nu_{n-q+\gamma}(dM)\nu_{q}(dL),$

and hence

$F^{\prime}(h)=\int_{G(n,q)}\int_{G(n,n-q+\gamma)}\int_{S^{n-1}\cap
M^{\perp}}f((M+hu)\cap L)h^{q-\gamma-1}\ \mathcal{H}^{q-\gamma-1}(dy)\
\nu_{n-q+\gamma}(dM)\nu_{q}(dL)$

for almost every $h>0$ due to the Lebesgue differentiation theorem. This
completes the proof of the lemma. ∎

######  Theorem 6.2.

Fix $n\geq 2$, $q\in\\{1,\ldots,n-1\\}$ and $\gamma\in\\{0,\ldots,q-1\\}$. Let
$L$ be a random linear subspace with distribution $\nu_{q}$ and let $E$ be a
stochastically independent random affine subspace, tangent to the unit sphere,
with distribution $\sigma_{n-q+\gamma}$ given by (6.1). Then the random
variable $d(o,E\cap L)^{-2}$ has a beta distribution with shape parameters

$a=\frac{\gamma+1}{2}\qquad\text{ and }\qquad b=\frac{n-q}{2}.$

More explicitly,

$\displaystyle
g_{n,q,\gamma}(r)={\omega_{\gamma+1}\omega_{n-q}\over\omega_{n-(q-\gamma)+1}}\,r^{-(\gamma+2)}\Big{(}1-\frac{1}{r^{2}}\Big{)}^{\frac{n-q}{2}-1}\,1_{\\{r>1\\}}$

is a probability density for $d(o,E\cap L)$.

Figure 6.1 illustrates the variety of density functions $g_{n,q,\gamma}$.

Figure 6.1: From left to right: The probability densities $g_{2,1,0}$,
$g_{3,1,0}$ and $g_{9,5,3}$.

###### Proof.

Let $X$ be a beta-distributed random variable on $(0,1)$ with shape parameters
$a,b>0$. Then $X^{-1/2}$ has tail probabilities

$\displaystyle 1-F_{X^{-1/2}}(t)$ $\displaystyle=\mathbb{P}\Big{[}X\leq{1\over
t^{2}}\Big{]}$
$\displaystyle=\frac{1}{B(a,b)}\int_{0}^{1/t^{2}}x^{a-1}(1-x)^{b-1}\ dx$
$\displaystyle=\frac{2}{B(a,b)}\int_{t}^{\infty}r^{-2a-1}\Big{(}1-\frac{1}{r^{2}}\Big{)}^{b-1}\
dr,\qquad t>1.$ (6.2)

On the other hand, applying Lemma 6.1 with $f(E)=1_{\\{d(o,E)\geq t\\}}$
yields

$\displaystyle\frac{1}{\omega_{q-\gamma}}\int_{G(n,q)}$
$\displaystyle\int_{G(n,n-q+\gamma)}\int_{S^{n-1}\cap
M^{\perp}}1_{\\{d(o,(M+hu)\cap L)\geq t\\}}\ \mathcal{H}^{q-\gamma-1}(du)\
\nu_{n-q+\gamma}(dM)\nu_{q}(dL)$
$\displaystyle=D(n,q,\gamma)\frac{\omega_{n-\gamma}}{\omega_{q-\gamma}}h^{\gamma+1}\int_{h}^{\infty}1_{\\{r\geq
t\\}}r^{-(\gamma+2)}\Big{(}1-\frac{h^{2}}{r^{2}}\Big{)}^{\frac{n-q}{2}-1}\ dr$
(6.3)

for almost all $h>0$. Using monotone convergence, one can show that both sides
of (6.3) are continuous in $h$, so this relation extends to all $h>0$. Letting
$h=1$, we see that the tail distributions of $d(o,E\cap L)$ are proportional
to

$\int_{t}^{\infty}r^{-(\gamma+2)}\Big{(}1-\frac{1}{r^{2}}\Big{)}^{\frac{n-q}{2}-1}\
dr,\qquad t>1.$

A comparison with (6.2) shows the first claim. The explicit density
$g_{n,q,\gamma}$ is obtained by differentiation of (6.3) when $h=1$. ∎

Let us remark in this context that random affine subspaces with the property
that their squared distance to the origin follows a beta distribution arise
naturally in the theory of random beta polytopes. For example, let
$X_{0},\ldots,X_{r}$ with $r\in\\{1,\ldots,n\\}$ be stochastically independent
random points in $B^{n}$ with probability density proportional to
$(1-\|x\|^{2})^{\nu-2\over 2}$ for some $\nu>0$, and consider the random
variable $d(o,M)^{2}$, where $M\in A(n,r)$ is the affine hull of
$X_{0},\ldots,X_{r}$. Then it follows from [10, Thm. 2.7] that this random
variable has the beta distribution $\operatorname{Beta}({n-r\over
2},{\nu(r+1)+r(n-1)\over 2})$.

From the explicit density of $d(o,E\cap L)$ in Theorem 6.2, the existence of
moments can directly be read off.

######  Corollary 6.3.

Let $\alpha\in\mathbb{R}$. In the set-up just introduced, one has that
$\mathbb{E}d(o,E\cap L)^{\alpha}<\infty$ if and only if $\alpha<\gamma+1$. In
particular, if $\gamma=0$ then the random variable $d(o,E\cap L)$ has infinite
expectation. If $\gamma\geq 1$ we have

$\mathbb{E}d(o,E\cap
L)={\omega_{\gamma+1}\omega_{n-q+\gamma}\over\omega_{\gamma}\omega_{n-(q-\gamma)+1}}=\frac{\omega_{n-q+\gamma}}{\omega_{\gamma}}(2\pi)^{-(n-q)}.$

Applying Stirling’s formula [1, §6.1.37] we conclude that for fixed
$\gamma\geq 1$,

$\frac{\mathbb{E}d(o,E\cap
L)}{\sqrt{n}}\longrightarrow{1\over\sqrt{2\pi}}{\omega_{\gamma+1}\over\omega_{\gamma}},$

as $n\to\infty$, independently of $q$.

##  7 Proofs of Theorem 4.1, Corollary 4.4 and Theorem 4.5

###### Proof of Theorem 4.1.

Given $E\in A(n,q)$, $q\in\\{0,\ldots,n\\}$, we will write
$\operatorname{lin}(E)$ for the linear subspace in $G(n,q)$ parallel to $E$.
In other words, $\operatorname{lin}(E)=E-x$ for all $x\in E$.

For the proof of Theorem 4.1, fix $n\geq 1$, $q\in\\{1,\ldots,n-1\\}$ and
$\gamma\in\\{0,\ldots,q-1\\}$, and let $f:A(n,\gamma)\to[0,\infty)$ be a
measurable function. Due to Lemma 3.1, we may assume without loss of
generality that $f$ is rotation invariant. The integral of interest is

$I=\int_{G(n,q)}\int_{A(n,n-q+\gamma)}f(E\cap L)H(E)\ \mu_{n-q+\gamma}(dE)\
\nu_{q}(dL).$

First note that $I$ is indeed well-defined as for $\nu_{q}$-almost all $L\in
G(n,q)$ and $\mu_{n-q+\gamma}$-almost all $E\in A(n,n-q+\gamma)$ we have
$\dim(E\cap L)=\gamma$. This follows from [22, Lem. 13.2.1], applied to $L$
and $L^{\prime}=\operatorname{lin}(E)$.

Define the function $g:A(n,q)\times A(n,n-q+\gamma)\to\mathbb{R}$ by

$\displaystyle g(E_{1},E_{2})=f\big{(}\operatorname{lin}(E_{1})\cap
E_{2}\big{)}1_{\\{d(o,E_{1})\leq 1\\}}H(E_{2}),$

where we let $f(\operatorname{lin}(E_{1})\cap E_{2})=0$ if
$\dim(\operatorname{lin}(E_{1})\cap E_{2})\neq\gamma$. Definition (2.3)
implies

$I=\frac{1}{\kappa_{n-q}}\int_{A(n,q)}\int_{A(n,n-q+\gamma)}g(E_{1},E_{2})\,\mu_{n-q+\gamma}(dE_{2})\mu_{q}(dE_{1}).$

Applying [22, Thm. 7.2.8] with $s_{1}=q$ and $s_{2}=n-q+\gamma$ yields

$\displaystyle
I=\frac{\bar{b}}{\kappa_{n-q}}\int_{A(n,\gamma)}\int_{A(E,q)}\int_{A(E,n-q+\gamma)}$
$\displaystyle
g(E_{1},E_{2})[\operatorname{lin}(E_{1}),\operatorname{lin}(E_{2})]^{\gamma+1}$
$\displaystyle\times\ \mu_{n-q+\gamma}^{E}(dE_{2})\ \mu_{q}^{E}(dE_{1})\
\mu_{\gamma}(dE),$ (7.1)

with the constant $\bar{b}$ given by

$\bar{b}=b_{n,n-\gamma}\frac{b_{n-\gamma,n-q}b_{n-\gamma,q-\gamma}}{b_{n,n-q}b_{n,q-\gamma}},\qquad
b_{i,j}={\omega_{i-j+1}\cdots\omega_{i}\over\omega_{1}\cdots\omega_{j}},\
i\in\mathbb{N},j\in\\{1,\ldots,i\\}.$ (7.2)

Expanding (7.1) by decomposing the measure $\mu_{\gamma}$ according to (2.3),
applying [22, Eq. (13.14)] and Tonelli’s theorem yields

$I=\frac{\bar{b}}{\kappa_{n-q}}\int_{G(n,\gamma)}I_{1}(L_{0},f)\
\nu_{\gamma}(dL_{0})$ (7.3)

with

$\displaystyle I_{1}(L_{0},f)$
$\displaystyle=\int_{G(L_{0},q)}\int_{G(L_{0},n-q+\gamma)}\int_{L_{0}^{\perp}}f((M+t)\cap
L)1_{\\{d(o,L+t)\leq 1\\}}H(M+t)$ $\displaystyle\hskip 113.81102pt\times\
\lambda_{L_{0}^{\perp}}(dt)[M,L]^{\gamma+1}\ \nu_{n-q+\gamma}^{L_{0}}(dM)\
\nu_{q}^{L_{0}}(dL).$

Since

$\int_{G(L_{0},p)}h(L)\
\nu_{p}^{L_{0}}(dL)=\int_{G(L_{0}^{\perp},p-\gamma)}h(L_{0}+L)\
\nu_{p-\gamma}^{L_{0}^{\perp}}(dL)$

for all $L_{0}\in G(n,\gamma)$ and for any measurable function
$h:G(L_{0},p)\to[0,\infty)$ and integers $\gamma<p<n$, we conclude

$\begin{split}I_{1}(L_{0},f)&=\int_{G(L_{0}^{\perp},q-\gamma)}\int_{G(L_{0}^{\perp},n-q)}\int_{L_{0}^{\perp}}f\Big{(}\big{(}(M+t)\cap
L\big{)}+L_{0}\Big{)}1_{\\{d(o,L+t)\leq 1\\}}\\\ &\qquad\qquad\times
H(M+L_{0}+t)\ \lambda_{L_{0}^{\perp}}(dt)[M,L]^{\gamma+1}\
\nu_{n-q}^{L_{0}^{\perp}}(dM)\ \nu_{q-\gamma}^{L_{0}^{\perp}}(dL),\end{split}$
(7.4)

where we also have used $(M+L_{0}+t)\cap(L+L_{0})=\big{(}(M+t)\cap
L\big{)}+L_{0}$, $d(o,L+L_{0}+t)=d(o,L+t)$ and $[M+L_{0},L+L_{0}]=[M,L]$.

Now consider (7.4) for fixed $L_{0}\in G(n,\gamma)$, $M\in
G(L_{0}^{\perp},n-q)$ and $L\in G(L_{0}^{\perp},q-\gamma)$. As $f$ and $H$ are
both rotation invariant, we may write

$\displaystyle f\big{(}((M+t)\cap L)+L_{0}\big{)}$
$\displaystyle=f_{I}\big{(}d(o,(M+t)\cap L)\big{)},$ $\displaystyle
H(M+L_{0}+t)$ $\displaystyle=H_{I}(\|t|M^{\perp}\|),$

with $f_{I}$ and $H_{I}$ satisfying (3.1). Thus, we may identify
$L_{0}^{\perp}$ with $\mathbb{R}^{n-\gamma}$ and conclude that

$\displaystyle I_{1}(L_{0},f)$
$\displaystyle=\int_{G(n-\gamma,q-\gamma)}\int_{G(n-\gamma,n-q)}I_{2}(L,M)[M,L]^{\gamma+1}\nu_{n-q}(dM)\
\nu_{q-\gamma}(dL),$ (7.5)

with

$\displaystyle I_{2}(M,L)$
$\displaystyle=\int_{\mathbb{R}^{n-\gamma}}f_{I}(d(o,(M+t)\cap
L))1_{\\{d(o,L+t)\leq 1\\}}H_{I}(\|t|M^{\perp}\|)\ \lambda_{{n-\gamma}}(dt)$
$\displaystyle=\int_{M}\int_{M^{\perp}}f_{I}(d(o,(M+x)\cap
L))H_{I}(\|x\|)1_{\\{d(o,L+x+y)\leq 1\\}}\ \ \lambda_{M^{\perp}}(dx)\
\lambda_{M}(dy)$

for $M\in G(n-\gamma,n-q)$ and $L\in G(n-\gamma,q-\gamma)$. Applying (2.4) to
the Lebesgue integral over $M^{\perp}$ we get

$\displaystyle I_{2}(M,L)$
$\displaystyle=[M,L]\int_{M}\int_{L}f_{I}(d(o,(M+t)\cap
L))H_{I}(\|t|M^{\perp}\|)1_{\\{d(o,L+t|M^{\perp}+y)\leq 1\\}}\ \
\lambda_{L}(dt)\ \lambda_{M}(dy)$ $\displaystyle=\int_{L}f_{I}(d(o,(M+t)\cap
L))H_{I}(\|t|M^{\perp}\|)I_{3}(M,L,t)\ \lambda_{L}(dt),$

where we first used the fact that $M+t|M^{\perp}=M+t$ holds for any $t\in L$,
and then Tonelli’s theorem. Here,

$I_{3}(M,L,t)=[M,L]\int_{M}1_{\\{\|(t|M^{\perp})|L^{\perp}+y|L^{\perp}\|\leq
1\\}}\ \lambda_{M}(dy).$

Another application of (2.4) reveals that

$I_{3}(M,L,t)=\int_{L^{\perp}}1_{\\{\|(t|M^{\perp})|L^{\perp}+z\|\leq 1\\}}\
\lambda_{L^{\perp}}(dz)=\kappa_{n-q}$

is the volume of a unit ball in $L^{\perp}$, centered at
$(t|M^{\perp})|L^{\perp}$. Inserting this into $I_{2}$ gives

$I_{2}(M,L)=\kappa_{n-q}\int_{L}f_{I}(d(o,(M+t)\cap L))H_{I}(\|t|M^{\perp}\|)\
\lambda_{L}(dt).$

Since $t\in L$ and $M\cap L=\\{o\\}$ for almost all $L$ and $M$, we have

$d(o,(M+t)\cap L)=d(o,\\{t\\})=\|t\|.$

This, and the use of spherical coordinates in $L$ give

$I_{2}(M,L)=\kappa_{n-q}\int_{0}^{\infty}f_{I}(r)r^{q-\gamma-1}\int_{S^{n-\gamma-1}\cap
L}H_{I}(r[u,M])\ \mathcal{H}^{q-\gamma-1}(du)\ dr.$

Inserting this into (7.5) and the result into (7.3), we get after an
application of Tonelli’s theorem that

$\displaystyle I$
$\displaystyle=\bar{b}\int_{0}^{\infty}f_{I}(r)r^{q-\gamma-1}\int_{G(n-\gamma,n-q)}J_{H}(M,r)\
\nu_{n-q}(dM)\ dr,$ (7.6)

with

$\displaystyle J_{H}(M,r)=\int_{G(n-\gamma,q-\gamma)}\int_{S^{n-\gamma-1}\cap
L}H_{I}(r[u,M])\mathcal{H}^{q-\gamma-1}(du)[M,L]^{\gamma+1}\
\nu_{q-\gamma}(dL).$

An invariance argument and [22, Thm. 7.1.1] imply

$\displaystyle
J_{H}(M,r)=\frac{\omega_{q-\gamma}}{\omega_{n-\gamma}}\int_{S^{n-\gamma-1}}H_{I}(r[u,M])\int_{G(\operatorname{span}u,q-\gamma)}[M,L]^{\gamma+1}\
\nu_{q-\gamma}^{\operatorname{span}u}(dL)\ \mathcal{H}^{n-\gamma-1}(du).$

Applying Lemma 3.2 in $\mathbb{R}^{n-\gamma}$ to the innermost integral in
$J_{H}(M,r)$ yields

$J_{H}(M,r)=c_{1}\int_{S^{n-\gamma-1}}H_{I}(r[u,M])[u,M]^{\gamma+1}\
\mathcal{H}^{n-\gamma-1}(du)$

with the constant
$c_{1}=a(n-\gamma,n-q,q-\gamma,\gamma+1)\frac{\omega_{q-\gamma}}{\omega_{n-\gamma}}$.
To simplify $J_{H}(M,r)$ further, we use the fact that $[u,M]=\|u|M^{\perp}\|$
and apply [2, Lem. 1] with $B_{p}=M^{\perp}$, $p=q-\gamma$ and $d=n-\gamma$.
This yields

$\displaystyle J_{H}(M,r)$
$\displaystyle=\frac{c_{1}}{2}\omega_{n-q}\omega_{q-\gamma}\int_{0}^{1}H_{I}(rt^{\frac{1}{2}})t^{\frac{q-1}{2}}(1-t)^{\frac{n-q}{2}-1}\
dt$
$\displaystyle=c_{1}\omega_{n-q}\omega_{q-\gamma}\int_{0}^{1}H_{I}(rz)z^{q}(1-z^{2})^{\frac{n-q}{2}-1}\
dz,$

using the substitution $z=\sqrt{t}$ in the last step. A comparison with (4.1)
gives $J_{H}(M,r)=c_{1}\omega_{n-q}\omega_{q-\gamma}J_{H}(r)$, so abbreviating

$c_{2}=\bar{b}c_{1}\frac{\omega_{n-q}\omega_{q-\gamma}}{\omega_{n-\gamma}},$

(7.6) becomes

$\displaystyle I$
$\displaystyle=c_{2}\omega_{n-\gamma}\int_{0}^{\infty}f_{I}(r)r^{q-\gamma-1}J_{H}(r)\
dr$ $\displaystyle=c_{2}\int_{A(n,\gamma)}f(E)d(o,E)^{-(n-q)}J_{H}(d(o,E))\
\mu_{\gamma}(dE),$

where we used (2.3) and spherical coordinates in $\operatorname{lin}(E)$.
Hence, the theorem is shown once we have confirmed that

$c_{2}=D(n,q,\gamma).$ (7.7)

We have

$c_{1}=\frac{\omega_{q-\gamma}}{\omega_{n-\gamma}}\prod_{i=1}^{n-q}\frac{\Gamma(\frac{n-\gamma-i}{2})\Gamma(\frac{n-q-i+\gamma}{2}+1)}{\Gamma(\frac{n-i+1}{2})\Gamma(\frac{n-q-i+1}{2})}=\frac{\omega_{q-\gamma}}{\omega_{n-\gamma}}\prod_{i=1}^{n-q}\frac{\omega_{n-i+1}\omega_{n-q-i+1}}{\omega_{n-\gamma-i}\omega_{n-q+\gamma+2-i}},$

so direct insertion gives

$\displaystyle\bar{b}c_{1}$
$\displaystyle=\Bigg{(}\frac{\omega_{\gamma+1}\cdots\omega_{n}}{\omega_{1}\cdots\omega_{n-\gamma}}\cdot\frac{\omega_{q-\gamma+1}\cdots\omega_{n-\gamma}}{\omega_{q+1}\cdots\omega_{n}}\cdot\frac{\omega_{n-q+1}\cdots\omega_{n-\gamma}}{\omega_{n-(q-\gamma)+1}\cdots\omega_{n}}\Bigg{)}$
$\displaystyle\hskip
113.81102pt\times\Bigg{(}\frac{\omega_{q+1}\cdots\omega_{n}}{\omega_{q-\gamma}\cdots\omega_{n-\gamma-1}}\cdot\frac{\omega_{1}\cdots\omega_{n-q}}{\omega_{\gamma+2}\cdots\omega_{n-(q-\gamma)+1}}\Bigg{)}\frac{\omega_{q-\gamma}}{\omega_{n-\gamma}}$
$\displaystyle=\Bigg{(}\frac{\omega_{q+1}\cdots\omega_{n}}{\omega_{q+1}\cdots\omega_{n}}\cdot\frac{\omega_{\gamma+1}\cdots\omega_{n}}{\omega_{\gamma+2}\cdots\omega_{n}}\cdot\frac{1}{\omega_{n-(q-\gamma)+1}}\Bigg{)}$
$\displaystyle\hskip
113.81102pt\times\Bigg{(}\frac{\omega_{q-\gamma}\cdots\omega_{n-\gamma}}{\omega_{q-\gamma}\cdots\omega_{n-\gamma}}\cdot\frac{\omega_{1}\cdots\omega_{n-\gamma}}{\omega_{1}\cdots\omega_{n-\gamma}}\Bigg{)}$
$\displaystyle=\frac{\omega_{\gamma+1}}{\omega_{n-(q-\gamma)+1}},$

where the products were suitably sorted at the second equality sign, and
simplified at the third equality sign. We thus get

$c_{2}=\frac{\omega_{\gamma+1}\omega_{q-\gamma}\omega_{n-q}}{\omega_{n-(q-\gamma)-1}\omega_{n-\gamma}}=D(n,q,\gamma).$

This shows (7.7) and completes the proof. ∎

###### Proof of Corollary 4.4.

For a Borel set $B\subseteq G(n,q)$, define

$\tilde{\nu}_{q}(B)=\int_{G(n,q_{1})}\cdots\int_{G(n,q_{\ell})}1_{B}(L_{1}\cap\ldots\cap
L_{\ell})\,\nu_{q_{\ell}}(dL_{\ell})\ldots\nu_{q_{1}}(dL_{1}).$

This gives rise to an invariant probability measure $\tilde{\nu}_{q}$ on
$G(n,q)$. However, by [22, Thm. 13.2.11] there is only one such measure, which
implies that $\tilde{\nu}_{q}=\nu_{q}$. This allows us in a first step to
reduce the outer integral over $G(n,q_{1}),\ldots,G(n,q_{m})$ in $I_{\ell,m}$
to a single integral over $G(n,q)$:

$\begin{split}I_{\ell,m}&=\int_{G(n,q)}\int_{A(n,p_{1})}\cdots\int_{A(n,p_{m})}f(E_{1}\cap\ldots\cap
E_{m}\cap L)\\\ &\qquad\qquad\qquad\qquad\times H(E_{1}\cap\ldots\cap
E_{m})\,\mu_{p_{m}}(dE_{m})\ldots\mu_{p_{1}}(dE_{1})\nu_{q}(dL).\end{split}$
(7.8)

To also convert the inner integrals over $A(n,p_{1}),\ldots,A(n,p_{m})$ to a
single integral, let $B$ be a Borel set in $A(n,n-q+\gamma)$ and define

$\displaystyle\widehat{\mu}_{n-q+\gamma}(B)=\int_{A(n,p_{1})}\cdots\int_{A(n,p_{m})}1_{B}(E_{1}\cap\ldots\cap
E_{m})\,\mu_{p_{m}}(dE_{m})\ldots\mu_{p_{1}}(dE_{1}).$

The measure $\widehat{\mu}_{n-q+\gamma}$ is motion invariant on
$A(n,n-q+\gamma)$. However, by [22, Thm. 13.1.3 and Thm. 13.2.12] all such
measures are constant multiples of the invariant measure $\mu_{n-q+\gamma}$,
so there exists some constant $c>0$ such that
$\widehat{\mu}_{n-q+\gamma}=c\mu_{n-q+\gamma}$. To determine the value of the
constant $c$, we employ a special case of the Crofton formula [22, Thm.
5.1.1]. It says that

$\displaystyle\int_{A(n,k)}\mathcal{H}^{i}(E\cap
W)\,\mu_{k}(dE)={\omega_{n+1}\omega_{i+1}\over\omega_{k+1}\omega_{n-k+i+1}}\mathcal{H}^{n-k+i}(W)$

for $0\leq i\leq k\leq n-1$ and where $W\subset\mathbb{R}^{n}$ is a convex set
of dimension $n-k+i$, that is, the affine hull of $W$ has dimension $n-k+i$.
Applying Crofton’s formula with $i=k=n-q+\gamma$ and $W=B^{n}$ we obtain

$\displaystyle\int_{A(n,n-q+\gamma)}\mathcal{H}^{n-q+\gamma}(E\cap
B^{n})\,\widehat{\mu}_{n-q+\gamma}(dE)$
$\displaystyle=c\int_{A(n,n-q+\gamma)}\mathcal{H}^{n-q+\gamma}(E\cap
B^{n})\,\mu_{n-q+\gamma}(dE)$
$\displaystyle=c\mathcal{H}^{n}(B^{n})=c\kappa_{n}.$

On the other hand, by applying Crofton’s formula repeatedly to each of the
integrals in its definition, the integral on the left is

$\displaystyle\int_{A(n,p_{1})}\cdots\int_{A(n,p_{m})}\mathcal{H}^{n-q+\gamma}(E_{1}\cap\ldots\cap
E_{m}\cap B^{n})\,\mu_{p_{m}}(dE_{m})\ldots\mu_{p_{1}}(dE_{1})$
$\displaystyle={\omega_{n+1}\omega_{n-q+\gamma+1}\over\omega_{p_{m}+1}\omega_{2n-p_{m}-q+\gamma+1}}\int_{A(n,p_{1})}\cdots\int_{A(n,p_{m-1})}\mathcal{H}^{2n-p_{m}-q+\gamma}(E_{1}\cap\ldots\cap
E_{m-1}\cap B^{n})$ $\displaystyle\hskip
227.62204pt\mu_{p_{m-1}}(dE_{m-1})\ldots\mu_{p_{1}}(dE_{1})$
$\displaystyle\vdots$
$\displaystyle={\omega_{n+1}\omega_{n-q+\gamma+1}\over\omega_{p_{m}+1}\omega_{2n-p_{m}-q+\gamma+1}}{\omega_{n+1}\omega_{2n-p_{m}-q+\gamma}\over\omega_{p_{m-1}}\omega_{3n-p_{m}-p_{m-1}-q+\gamma+1}}\cdots$
$\displaystyle\hskip 170.71652pt\cdots{\omega_{n+1}\omega_{mn-p_{m}-\ldots-
p_{2}-q+\gamma}\over\omega_{p_{1}+1}\omega_{n-p_{1}+mn-p_{m}-\ldots-
p_{2}-q+\gamma+1}}\,\mathcal{H}^{n}(B^{n})$
$\displaystyle={\omega_{n+1}^{m}\omega_{n-q+\gamma+1}\over\omega_{p_{1}}\cdots\omega_{p_{m}}\omega_{n+1}}\,\kappa_{n}.$

Here, we simplified the telescopic product of the $\omega$-terms and used our
assumption $p_{1}+\ldots+p_{m}-(m-1)n=n-1+\gamma$ in the last step. A
comparison of these two expressions implies

$\widehat{\mu}_{n-q+\gamma}={\omega_{n+1}^{m}\omega_{n-q+\gamma+1}\over\omega_{p_{1}}\cdots\omega_{p_{m}}\omega_{n+1}}\,\mu_{n-q+\gamma}.$

As a consequence, we can reduce the inner integrals in (7.8) to a single
integral over $A(n,n-q+\gamma)$:

$\displaystyle
I_{\ell,m}={\omega_{n+1}^{m}\omega_{n-q+\gamma+1}\over\omega_{p_{1}}\cdots\omega_{p_{m}}\omega_{n+1}}\int_{G(n,q)}\int_{A(n,n-q+\gamma)}f(E\cap
L)H(E)\,\mu_{n-q+\gamma}(dE)\nu_{q}(dL).$

The result of Corollary 4.4 can now be concluded from Theorem 4.1. ∎

The proof that the results of Theorems 4.1 and 4.5 are equivalent, is based on
the following Blaschke–Petkantschin formula:

$\int_{A(n,k)}g(E)\mu_{k}(dE)=\frac{\omega_{n-k}}{\omega_{r-k}}\int_{G(n,r)}\int_{A(L,k)}g(E)d(o,E)^{n-r}\mu_{k}^{L}(dE)\nu_{r}(dL),$
(7.9)

with integers $0\leq k<r<n$ and measurable $g:A(n,k)\to[0,\infty)$, see e.g.
[14, p. 33].

###### Proof of the equivalence of Theorems 4.1 and 4.5.

That Theorem 4.5 implies Theorem 4.1 follows rather directly by invariant
integration of (4.4) with respect to $L_{0}$ and an application of the
Blaschke–Petkantschin formula (7.9) with $r=q$ and $k=\gamma$.

Assume now that Theorem 4.1 holds, and let $f:A(L_{0},\gamma)\to[0,\infty)$ be
measurable. Assume first that $f$ is rotation invariant under all rotations
fixing $L_{0}$. The function $\tilde{f}(E)=f_{I}(d(o,E))$, $E\in A(n,\gamma)$,
is the rotation invariant extension of $f$ to $A(n,\gamma)$. Hence, the
invariance of $\mu_{n-q+\gamma}$ gives

$\displaystyle\int_{G(n,q)}\int_{A(n,n-q+\gamma)}\tilde{f}(E\cap L)H(E)\
\mu_{n-q+\gamma}(dE)\ \nu_{q}(dL)$
$\displaystyle=\int_{SO(n)}\int_{A(n,n-q+\gamma)}\tilde{f}(E\cap\vartheta
L_{0})H(E)\ \mu_{n-q+\gamma}(dE)\ \nu(d\vartheta)$
$\displaystyle=\int_{SO(n)}\int_{A(n,n-q+\gamma)}\tilde{f}\big{(}\vartheta(E\cap
L_{0})\big{)}H(\vartheta^{-1}E)\ \mu_{n-q+\gamma}(dE)\ \nu(d\vartheta)$
$\displaystyle=\int_{A(n,n-q+\gamma)}f(E\cap L_{0})H(E)\
\mu_{n-q+\gamma}(dE).$

On the other hand, the Blaschke–Petkantschin relation (7.9) and a similar
reasoning shows

$\int_{A(n,\gamma)}\tilde{f}(E)d(o,E)^{-(n-q)}J_{H}(d(o,E))\
\mu_{\gamma}(dE)=\frac{\omega_{n-\gamma}}{\omega_{q-\gamma}}\int_{A(L_{0},\gamma)}f(E)J_{H}(d(o,E))\
\mu_{\gamma}^{L_{0}}(dE).$

Theorem 4.1 states that these two displayed expressions coincide up to
multiplication with the constant $D(n,q,\gamma)$, and thus (4.4) holds for the
function $f$ chosen.

With arguments as in the proof of Lemma 3.1, one can show that (4.4) holds for
all measurable $f:A(L_{0},\gamma)\to[0,\infty)$ if it holds for all such
functions which are in addition invariant under all rotations fixing $L_{0}$.
This proves that Theorem 4.5 holds and concludes the proof of the equivalence.
∎

### Acknowledgment

ED has been supported by Sapere Aude: DFF-Starting Grant 0165-00061B. CT has
been supported by the German Research Foundation (DFG) via SPP 2265 Random
Geometric Systems and the project Limit theorems for the volume of random
projections of $\ell_{p}$-balls (project number 516672205).

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