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% Author Package file for use with AMS-LaTeX 1.2
\controldates{18-NOV-1998,18-NOV-1998,18-NOV-1998,18-NOV-1998}
 
\documentclass{era-l}
\usepackage[cmtip,all]{xy}


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%\newcommand{\g}{{B @<{\pi_1^2}<< A_2 @>{\pi_2^2}>> \Gamma_2}}
\newcommand{\g}{{B \xleftarrow{\pi_1^2} A_2 \xrightarrow{\pi_2^2}\Gamma_2}}
%\newcommand{\h}{{B @<{\pi_1^3}<< A_3 @>{\pi_2^3}>> \Gamma_3}}
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\newtheorem{thm}{Theorem}[section]
\newtheorem{cor}{Corollary}[section]
\newtheorem{lemma}{Lemma}[section]
\newtheorem{prop}{Proposition}[section]
\newtheorem{defiprop}{Proposition-Definition}[section]

\theoremstyle{definition}
\newtheorem{defi}{Definition}[section]
\newtheorem*{examn}{Example}

\theoremstyle{remark}
\newtheorem*{remks}{Remarks}

%%%%%%%%%%%%%%%%%
\begin{document}%
%%%%%%%%%%%%%%%%%

\title{Crofton formulas in projective Finsler spaces}


%%%%%%%%%%%%%%%%%%%%%%%%%%JUAN CARLOS%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\author{J. C. \'Alvarez Paiva}
\address{Universit\'e Catholique de Louvain, 
Institut de Math\'ematique Pure et Appl., Chemin du Cyclotron 2, B-1348
Louvain-la-Neuve, Belgium}
\email{alvarez@agel.ucl.ac.be}
%%%%%%%%%%%%%%%%%%%%%%%%EMMANUEL%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\author{E. Fernandes}
\address{Universit\'e Catholique de Louvain, Institut
de Math\'ematique Pure et Appl., Chemin du Cyclotron 2, B-1348
Louvain-la-Neuve, Belgium}
\email{fernandes@agel.ucl.ac.be}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%    General info
\subjclass{Primary 53C65; Secondary 53C60}

\issueinfo{4}{13}{}{1998}
\dateposted{November 23, 1998}
\pagespan{91}{100}
\PII{S 1079-6762(98)00053-5}
\def\copyrightyear{1998}
\copyrightinfo{1998}{American Mathematical Society}

\date{August 08, 1998}

\commby{Dmitri Burago}
\keywords{Crofton formulas, Hilbert's fourth problem, Finsler 
geometry}

%%%%%%%%%%%%%%%%%%%%%%ABSTRACT%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{abstract}
We extend the classical Crofton formulas in Euclidean integral geometry 
to Finsler metrics on $\R^n$ whose geodesics are straight lines.
\end{abstract}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
        
\thanks{Partially supported by a
       {\itshape credit aux chercheurs\/} from the FNRS}

\maketitle


%%%%%%%%%%%%%%%%%%%%%%%%%INTRODUCTION%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Introduction}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Hilbert's fourth problem asks to construct and study all
metric structures on convex subsets of $\R P^n$ such that
the straight line segment is the shortest curve joining two points.
Hilbert's motivation is given in his presentation of the problem
(see \cite{Hilbert}): his close friend Minkowski had defined what we now call 
finite-dimensional Banach spaces and Hilbert himself had modified
the Cayley-Klein construction of hyperbolic geometry to yield a
family of metric spaces where the straight lines are
geodesics. Moreover, Hilbert believed that the study of metric spaces
where the geodesics are straight lines ``will throw a new light upon
the idea of distance, as well as upon other elementary ideas.''

 
Busemann, who called the metrics appearing in Hilbert's fourth problem
{\itshape projective metrics\/}, proposed an integral-geometric construction 
which is inspiringly simple. Here is his construction 
of projective metrics on $\R^n$ (see \cite{Busemann:1}):

 Let $\Phi$ be a smooth (possibly signed) measure on the space of
hyperplanes such that if $x, y$, and $z$ are three noncollinear points, 
then the measure of the set of hyperplanes intersecting the wedge formed 
by the segments $xy$ and $yz$ is strictly positive. The $\Phi$-distance 
between two points is defined as one-half the measure of all hyperplanes 
intersecting the line segment joining them. 


Explicit formulas can be given if the space of  cooriented hyperplanes in 
$\R^n$ is identified with the manifold $S^{n-1} \times \R$ by assigning 
to each pair $(\xi,r) \in S^{n-1} \times \R$ the hyperplane 
$\{x \in \R^n : r = \xi \cdot x \}$ cooriented by $\xi$.  
The measure on  $S^{n-1} \times \R$  which descends to the measure $\Phi$ 
on the space of hyperplanes without coorientation can be written as 
$\nu |\Omega \wedge dr |$, where $\nu$ is a smooth function  and $\Omega$ is 
the standard volume form on $S^{n-1}$. If we define the function 
$L : T\R^n \rightarrow \R$ by the formula
\[
L(x,v)= {1\over 4} \int_{\xi\in S^{n-1}}|\xi\cdot v|\nu(\xi, \xi\cdot x)\Omega ,
\]
then the $\Phi$-distance between any two points $x$ and $y$ is given
as the infimum of the numbers $\int L({\dot \gamma}(t)) dt$, where 
$\gamma$ ranges over all smooth curves joining $x$ and $y$.

\begin{examn}
Taking $\nu : S^{1} \times \R \rightarrow \R$ to be the 
function $\nu(\theta,r) :=  1 + r^{2}$, we obtain
\[
L(x_{1}, x_{2}, v_{1}, v_{2}) = 
{1 \over 3\sqrt{ v_{1}^{2} +  v_{2}^{2}}} \cdot
(2x_{1} x_{2} v_{1} v_{2} + (3 + 2x_{1}^{2} +x_{2}^{2}) v_{1}^{2} +
(3 + 2x_{2}^{2} +x_{1}^{2}) v_{2}^{2}) .
\]

Remarkably, for any smooth projective metric there exists a smooth
(possibly signed) measure on the space of hyperplanes such that
the length of any line segment equals the measure of the set of
hyperplanes intersecting it
(see \cite{Pogorelov,Szabo} and \cite{Alvarez-Gelfand-Smirnov}).

{\itshape Do there exist measures $\Phi_{n-k}$, $1 \leq k \leq n-1$, on
the space of $(n-k)$-flats such that  the volume of any domain 
contained in a $k$-flat equals the measure of all 
$(n-k)$-flats intersecting it?}

In the case of finite-dimensional Banach spaces this question
is due to Busemann (see \cite{Busemann:2}). To make sense of it we 
need  to define the volume of a $k$-dimensional submanifold in a projective 
metric space. The definition will depend solely on the fact that the function 
$L$ is a {\itshape Finsler metric\/} on $\R^n$ (see 
\cite{Alvarez-Gelfand-Smirnov}):
\end{examn}
\begin{defi}
Let $M$ be a manifold and let $TM\backslash 0$ denote its tangent
bundle with the zero section deleted. A Finsler metric on $M$ is a smooth 
positive function
\[
L:TM\backslash 0\to\R
\]
with the following properties :
\begin{itemize}
\item $L(t v_m)=|t| L(v_m)$, for any real number
$t$ and any nonzero vector $v_m\in T_mM$.
\item For each $m\in M$, the set $\{v_m\in T_mM\,|\,L(v_m)=1\}\subset
T_mM$ is a smooth quadratically convex hypersurface.
\end{itemize}
\end{defi}

Note that the restriction of a Finsler metric to the tangent space of
any submanifold defines a Finsler metric on the submanifold. 

In order to define the volume of a Finsler manifold $(M,L)$, let 
$D^{*}_{m} M \subset T^{*}_{m}M$  be the dual to the convex set 
$\{v_m\in T_mM : L(v_m) <1 \} \subset T_mM$ and define the
{\itshape unit co-disc bundle\/}, $D^{*}M \subset T^{*}M$, as the the 
union of all  the $D^{*}_{m} M$, $m \in M$. 

\begin{defi}
The {\itshape volume\/} of an $n$-dimensional Finsler manifold $(M,L)$ is 
the symplectic volume of its unit co-disc bundle divided by the  volume 
of the Euclidean $n$-dimensional unit ball. The $k$-volume of a $k$-dimensional 
submanifold is the volume of the submanifold with its induced Finsler metric.
\end{defi} 

 We are now in a position to announce our main result:

\begin{thm}[Crofton formulas for projective Finsler spaces] 
\label{Crofton}
Let $L$ be a Finsler metric on $\R^n$ whose geodesics are straight 
lines and let $k$, $1 \leq k \leq n-1$, be a natural number. There
exists a smooth (possibly signed) measure $\Phi_{n-k}$ on the manifold 
$H_{n,n-k}$ of $(n-k)$-flats  such that if $N \subset \R^n$ is an 
immersed $k$-dimensional submanifold, then 
\begin{equation}\label{eq:1}
\mbox{\textit{vol}}_{k}(N) =  {1 \over \epsilon_{k}} \int_{\lambda\in H_{n,n-k}} 
\# (N\cap\lambda)\Phi_{n-k} ,
\end{equation}
where $\epsilon_{k}$ is the volume of the Euclidean unit ball of dimension $k$.
\end{thm}


To construct the  measures $\Phi_{n-k}$, $ 1 \leq k < n-1$, from the 
measure  $\Phi_{n-1}$, consider the fibration
\[
(\underbrace{H_{n,n-1}\times \cdots \times H_{n,n-1}}_{k \textup{ times}})
\backslash\Delta \xrightarrow{\pi} H_{n,n-k} ,
\]
where
\[
\Delta=\{(\lambda_{1},\dots, \lambda_{n-k})\in (H_{n,n-1})^{k} :
\operatorname{dim}(\lambda_{1} \cap \cdots \cap \lambda_{n-k}) > k \},
\]  
and
\[
\pi(\lambda_{1}, \dots, \lambda_{n-k}) := \lambda_{1} \cap \cdots \cap 
\lambda_{n-k} .
\]
We define $\Phi_{n-k}$ as the pushforward of the product measure on  
$(H_{n,n-1})^k \setminus \Delta$.

 This definition of the measures $\Phi_{n-k}$, together with the 
Crofton formulas and an easy application of Fubini's theorem, yields
the following result:

\begin{thm}\label{thm:2}
Let $L$ be a Finsler metric on $\R^n$ whose geodesics are straight 
lines and let $K$ be a smooth compact convex hypersurface in $\R^n$. 
For any $\lambda \in H_{n,n-k}$, $1 \leq k \leq  n-1$, let 
$\mbox{\textit{vol}}_{(n-k-1)}(K\cap\lambda)$ 
denote the $(n-k-1)$-volume of  $K\cap\lambda$. If $\Phi_{n-k}$ is the 
volume density of $H_{n,n-k}$ appearing in the Crofton formula, then 
there exists a constant $c$, independent of $K$, such that 
\begin{equation}
\mbox{\textit{vol}}_{(n-1)}(K) = c \cdot \int_{\lambda\in 
H_{n,n-k}} \mbox{\textit{vol}}_{(n-k-1)}(K\cap\lambda)\Phi_{n-k} .
\end{equation}
\end{thm} 

The Crofton formula \eqref{eq:1} was proved in the case of hypermetric, 
finite-dimensional Banach spaces by Schneider and Wieacker (see 
\cite{Schneider-Wieacker}). This is precisely the case where the
associated measure on the space of hyperplanes is nonnegative. 
It is likely that the proof in \cite{Schneider-Wieacker} can be extended 
to cover all finite-dimensional Banach spaces. We remark that our proof, 
which involves the symplectic structure  on the space of geodesics of a 
projective Finsler space, only covers those Banach spaces whose unit spheres 
are smooth and quadratically convex.


\begin{remks}
1. The volume of a Finsler manifold given here is 
based on the Holmes-Thompson volume of submanifolds of 
finite-dimensional Banach spaces (see \cite{Holmes-Thompson} and 
\cite{Thompson}).

2. Trivial modifications of Theorems \ref{Crofton} and \ref{thm:2} hold for  
projective metrics on any open convex subset of $\R P^n$, including
$\R P^n$ itself. In this announcement we have restricted our considerations
to $\R^n$ in
order to give shorter and clearer statements.
\end{remks}
 
 The first author is happy to acknowledge many fruitful discussions
with I.~M.~Gel\-fand. 



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Crofton formula for double fibrations}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

 
In \cite{Gelfand-Smirnov} Gelfand and Smirnov gave a first step in 
establishing the relation between the two main currents in integral 
geometry: the integral geometry of the Blaschke school and that of the 
Gelfand school. The main result of the present section can be taken as a 
point of departure in the simplification and extension of their work. 

Let us begin by recalling some preliminary notions: double fibrations
and densities.

\begin{defi}
A double fibration is a diagram of manifolds
\begin{equation*}
\begin{xy}
\xymatrix{
      &  A \ar[dl]_{\pi_1}\ar[dr]^{\pi_2} &   \\
      B  &     &  \Gamma}
\end{xy}
\end{equation*}
with the following properties:
\begin{itemize}
\item $\pi_{1}:A\to B$, $\pi_2:A\to\Gamma$ are fiber bundles.
\item The map $\pi_1\times\pi_2: A\to B\times\Gamma$ is a smooth
embedding.
\item For each $b \in B$ and for each $\gamma \in \Gamma$, the sets
$B_{\gamma} := \pi_{1}(\pi_{2}^{-1}(\gamma)) \subset B$ and 
$\Gamma_{b} := \pi_{2}(\pi_{1}^{-1}(b)) \subset \Gamma$ are 
smooth submanifolds.
\end{itemize}
\end{defi}




\begin{defi}
A $k$-\textit{density} $\varphi$ on a smooth manifold $M$ is a real-valued 
function of a point $x\in N$ and of $k$ linearly independent 
vectors $v=(v_1,\dots,v_k)$ in $T_xM$ such that, if $v'=(v_1',\dots,v_k')$
is another set of $r$ linearly independent vectors which generates the
same $k$-dimensional subspace of $T_{x}M$, i.e.,
$v'=Av$ with $A\in GL(k,\R)$, then
\[
\varphi(x;v')=|\det A|\varphi(x;v).
\]

The absolute value of a differential form of degree $k$ is a $k$-density. 
A Finsler metric is a $1$-density.
\end{defi}

 Because  densities change as the absolute value of the 
determinant, they can be integrated over unoriented and nonorientable
manifolds. Moreover, they can be pulled back under smooth maps in 
exactly the same way as differential forms.

To any double fibration $\db$ one  associates a number of 
integral transforms which send functions, differential forms, or 
densities on $\Gamma$ to similar objects on $B$. The definition of
the transforms is surprisingly simple:

Take a density (or form) $\Phi$ on $\Gamma$, pull it back to $A$ and 
push it forward to $B$ by fiber integration. 

Remark that the fiber integration of a form or density is not always defined.
In the rest of the paper we shall make the implicit assumption that the
fibers of $\pi_{1} : A \rightarrow B$ are compact and, when 
working with forms, that the fibers are oriented.

\begin{defiprop}
Let $B \xleftarrow{\pi_1} A \xrightarrow{\pi_2}
\Gamma$ be a double fibration. Let
$k$ be the dimension of the fibers of $\pi_{1} : A \rightarrow B$ and let
$\Phi$ be an $m$-density (resp. $m$-form) on $\Gamma$ with $m \geq k$.
The Gelfand transform of $\Phi$ is the $(m-k)$-density (resp. 
$(m-k)$-form) $\pi_{1*}\pi_2^*\Phi$.
\end{defiprop}

Classical integral-geometric transforms such as those of Radon, John, 
and Funk are particular cases of this construction. 

 We now describe a second construction involving double fibrations and
densities.
 
Let $\db$ be a double fibration with $\operatorname{dim}(B) = n$ and $\operatorname{dim}(B_{\gamma}) = 
n-k$. If $\Phi$ is a top-order density on $\Gamma$, we define
the following functional on the space of $k$-dimensional submanifolds
of $B$:
\[
S_{\Phi}(N) := \int_{\gamma \in \Gamma} \#(B_{\gamma} \cap N) \Phi .
\]

 These functionals were first considered by Busemann (see 
 \cite{Busemann:2} and \cite{Busemann:3})
in the particular case of the double fibration associated to $(n-k)$-flats in
$\R^n$. Also in this particular case, Gelfand and Smirnov proved that 
for any top-order density $\Phi$ on the space of $(n-k)$-flats there exists 
a $k$-density $\varphi$  on $\R^n$ such that
\[
\int_{\lambda \in H_{n,n-k}} \#(\lambda \cap N) \Phi = \int_{N} \varphi .
\]
Moreover, they showed through explicit formulas that the assignment
$\Phi \mapsto \varphi$ is an integral-geometric transform in the sense
of Radon, John, and Gelfand. Underlying these formulas is the following
general, and apparently new, result:
 
\begin{thm}[Crofton formula for double fibrations]
Let $\db$ be a \linebreak 
double fibration with $\operatorname{dim}(B) = n$ and $\operatorname{dim}(B_{\gamma}) = 
n-k$. If $\Phi$ is a top-order density on $\Gamma$ and $N \subset B$ is
a $k$-dimensional submanifold, then
\begin{equation}
S_{\Phi}(N)=\int_{\Gamma}\#(N\cap B_{\gamma})\Phi=\int_N\pi_{1*}\pi_2^*\Phi.
\end{equation}
\end{thm}

 Functorial properties of the Gelfand transform play an important 
role in the proof of the main theorem.

\begin{defi}
A morphism between two double fibrations 
$B \xleftarrow{\pi_1} A \xrightarrow{\pi_2} \Gamma$ and 
$B'\xleftarrow{\pi_1^\prime} A' \xrightarrow{\pi_2^\prime}
 \Gamma'$ is a commutative
diagram of fibrations
\begin{equation*}
\begin{xy}
\xymatrix{
      B \ar[d]_{\rho_{B}} & A \ar[d]_{\rho_{A}} \ar[l]_{\pi_{1}} 
      \ar[r]^{\pi_{2}} & \Gamma \ar[d]_{\rho_{\Gamma}}  \\
      B' & A'  \ar[r]^{\pi_{2}'} \ar[l]_{\pi_{1}'}  &  \Gamma' }
\end{xy}
\end{equation*}
\end{defi}

\begin{thm}
Let 
\begin{equation*}
\begin{xy}
\xymatrix{
      B \ar[d]_{\rho_{B}}  & A \ar[d]_{\rho_{A}} \ar[l]_{\pi_{1}}
      \ar[r]^{\pi_{2}} & \Gamma \ar[d]_{\rho_{\Gamma}}  \\
      B'  & A'  \ar[r]^{\pi_{2}'} \ar[l]_{\pi_{1}'} &  \Gamma'}
\end{xy}
\end{equation*}
be a morphism of double fibrations such that for every point $a' \in 
A'$ the map $\pi_{2}$ restricted to $\rho_{A}^{-1}(a')$ is a 
diffeomorphism onto $\rho_{\Gamma}^{-1}(\pi_{2}'(a'))$. If $\Phi$
is a density or differential form on $\Gamma$, then 
\[
\rho_{B*}\pi_{1*}\pi_{2}^{*} \Phi = \pi_{1*}' (\pi_{2}')^{*} 
\rho_{\Gamma *}\Phi .
\]
\end{thm}

\begin{thm}
Let $\f,\g$, and $\h$ be double fibrations and let 
$\rho:\Gamma_1\times\Gamma_2\to\Gamma_3$ be a fibration satisfying 
the following condition: 

The points $\gamma_{1} \in \Gamma_{1}$ and  $\gamma_{2} \in \Gamma_{2}$ 
are incident to $b \in B$ if and only if 
$\rho(\gamma_{1},\gamma_{2}) \in \Gamma_{3}$ is incident to $b$.

If $\Omega_1$ and $\Omega_2$  are top-order forms on $\Gamma_1$ 
and $\Gamma_2$ and if $p_1:\Gamma_1\times\Gamma_2\to\Gamma_1$, 
$p_2:\Gamma_1\times\Gamma_2\to \Gamma_2$ are the canonical projections,
then 
\[
(\pi_1^3)_*(\pi_2^3)^* \rho_*(p_1^*\Omega_1 \wedge p_2^*\Omega_2) 
=(\pi_1^1)_*(\pi_2^1)^*\Omega_1\wedge(\pi_1^2)_* (\pi_2^2)^*\Omega_2 .
\]
\end{thm}



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Integral geometry and symplectic geometry}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Let $(M,L)$ be a Finsler manifold such that its space of oriented
geodesics is a manifold $G(M)$. Let $S^{*}M$ denote the {\itshape unit
co-sphere bundle\/} of $M$ and let  $\pi : S^{*}M \rightarrow G(M)$ be the
canonical projection which sends a given unit covector to the geodesic 
which has this covector as initial condition. 

\begin{defiprop}[\cite{Arnold} and \cite{Besse}]
Let $(M,L)$ be a Finsler manifold whose space of oriented geodesics, 
$G(M)$, is a smooth manifold and let
\begin{equation*}
\begin{xy}
\xymatrix{
      S^{*}M \ar[d]_{\pi}\ar[r]^{i} & T^{*}M  \\
   G(M)                        &          }
\end{xy}
\end{equation*}
be the canonical projection onto $G(M)$ and the canonical inclusion
into $T^{*}M$. If $\omega_{0}$ is the standard symplectic form on
$T^{*}M$, then there is a unique symplectic form $\omega$ on $G(M)$
which satisfies the equation $\pi^{*}\omega = i^{*}\omega_{0}$.
\end{defiprop}


The case $k = n-1$, that of the volume of hypersurfaces, in 
Theorem~\ref{Crofton} follows from a more general result:

\begin{thm}[\cite{Alvarez:1}] \label{general}
Let $M$ be an $n$-dimensional Finsler manifold whose space of oriented 
geodesics, $G(M)$, is a smooth manifold. If $N \subset M$ is an immersed 
hypersurface and if $\omega^{n-1}$ denotes the Liouville volume form on $G(M)$, 
then 
\[
\mbox{\textit{vol}}(N) = {1 \over 2\epsilon_{n-1}} \cdot \int_{\gamma \in G(M)} 
\#(\gamma \cap N) |\omega^{n-1}| ,
\]
where $\epsilon_{n-1}$ is the volume of the Euclidean unit ball
of dimension $n-1$.
\end{thm}


The symplectic structure on the space of oriented straight lines, 
$H_{n,1}^{+}$, induced by a projective Finsler metric on $\R^n$ has
a simple characterization: 

\begin{thm}[\cite{Alvarez:2}] \label{characterization}
Let $L$ be a projective Finsler metric on $\R^n$ and let $\omega$ 
be the symplectic form on the space of oriented lines induced by $L$. 
The $2$-form $\omega$ is characterized, up to sign, by the following
properties:
\begin{itemize}
\item The form $\omega$ is closed.
\item The form $\omega$ is odd: if $a : H_{n,1}^{+} \rightarrow H_{n,1}^{+}$
      denotes the involution that changes the orientation of the 
      lines, then $a^{*}\omega = -\omega$.
\item For any point $x \in \R^n$, the pull-back of $\omega$ to the 
      submanifold of all oriented lines passing through $x$ is 
      identically zero.
\item If $x$ and $y$ are two distinct points in $\R^n$ and $\Pi$ is any 
      $2$-flat containing them, then the integral of $|\omega|$ over 
      the set of all oriented lines lying on $\Pi$ and intersecting the
      segment $xy$ equals four times the distance between $x$ and $y$.                 
\end{itemize}
\end{thm}

Recall from the introduction that if $L$ is a  projective Finsler metric 
on $\R^n$, then it is given by the formula
\[
L(x,v)= {1\over 4} \int_{\xi\in S^{n-1}}|\xi\cdot v|\nu(\xi, \xi\cdot x)\Omega ,
\]
for some smooth function $\nu$ on $S^{n-1} \times \R$, which we 
identify with the space of oriented hyperplanes $H_{n,n-1}^{+}$.

 In order to write the symplectic form $\omega$ in terms on $\nu$, we
consider the form $\Omega_{n-1} := \nu \Omega \wedge dr$ and the 
double fibration
\begin{equation*}
\begin{xy}
\xymatrix{
      &  A \ar[dl]_{\pi_1}\ar[dr]^{\pi_2} &   \\
      H_{n,1}^{+}  &     &  H_{n,n-1}^{+} }
\end{xy}
\end{equation*}
where $A$ is the set 
$\{(l,\lambda) \in H_{n,1}^{+} \times H_{n,n-1}^{+} : l \subset \lambda \}$.
We remark that the fibers of this double fibration inherit a natural 
orientation from the orientation of $\R^n$ and those of the subspaces
involved and that it makes sense to compute the form
$\pi_{1*} \pi_{2}^{*} \Omega_{n-1}$. It follows from 
Theorem~\ref{characterization} that 
$\pi_{1*} \pi_{2}^{*} \Omega_{n-1} = \omega$.

\begin{thm} \label{Omega-omega}
The Gelfand transform of $\Omega_{n-1} := \nu \Omega \wedge dr$
associated to the double fibration 
$H^+_{n,1} \xleftarrow{\pi_1} A \xrightarrow{\pi_2} H^+_{n,n-1}$ equals the 
symplectic form induced by the Finsler metric $L$.
\end{thm}



We shall now use the form $\Omega_{n-1}$ to construct top-order forms
$\Omega_{n-k}$ on the spaces of oriented $(n-k)$-flats, 
$H_{n,n-k}^{+}$. To do this consider the fibration
\[
(\underbrace{H_{n,n-1}^{+}\times \cdots \times H_{n,n-1}^{+}}_{k 
\textup{ times}})
\backslash\Delta \xrightarrow\pi H_{n,n-k}^{+} ,
\]
where
\[
\Delta=\{(\lambda_{1},\dots, \lambda_{n-k})\in H_{n,n-1}\times 
\cdots  \times H_{n,n-1}\,|\, 
\operatorname{dim}(\lambda_{1} \cap \cdots \cap \lambda_{n-k}) > k \},
\]  
and
\[
\pi(\lambda_{1}, \dots, \lambda_{n-k}) := \lambda_{1} \cap \cdots \cap 
\lambda_{n-k} .
\]
The orientation of the intersection is given by that of $\R^n$ and 
those of the hyperplanes $\lambda_{i}, 1 \leq i \leq k$. Since this 
fibration is oriented, we may define the push-forward of forms 
and set $\Omega_{n-k} := \pi_{*}\Omega_{n-1}^{k}$.


 Let us also consider the double fibration 
\begin{equation*}
\begin{xy}
\xymatrix{
      &  A \ar[dl]_{\pi_1}\ar[dr]^{\pi_2} &   \\
      H_{n,1}^{+}  &     &  H_{n,n-k}^{+} }
\end{xy}
\end{equation*}
where $H_{n,n-k}^{+}$ denotes the space of oriented $(n-k)$-flats and
$A$ is the set 
$\{(l,\zeta) \in H_{n,1}^{+} \times H_{n,n-k}^{+} : l \subset \zeta \}$.
Like the case $k = 1$, the fibers are oriented and it makes sense to 
compute the form $\pi_{1*} \pi_{2}^{*} \Omega_{n-k}$. 

 The previous theorem together with the functorial properties of the Gelfand
transform imply the following

\begin{thm} \label{powers-of-omega}
The Gelfand transform of $\Omega_{n-k}$ associated 
to the double fibration 
$H^+_{n,1} \xleftarrow{\pi_1} A \xrightarrow{\pi_2} H^+_{n,n-k}$
equals the $2k$-form $\omega^{k}$. In particular, $\Omega_{1} = 
\omega^{n-1}$.
\end{thm}

 Let us clarify the relation between the forms $\Omega_{n-k}$ and the
measures $\Phi_{n-k}$ defined in the introduction.

Note that if $M$ is an oriented manifold of dimension $n$, there is a simple
correspondence between differential $n$-forms and $n$-densities:
if $\Psi$ is an $n$-form on $M$, we define an  $n$-density 
$\Phi$ also on $M$ by letting 
$\Phi(x,v_{1}, \dots , v_{n}) = \Psi_{x}(v_{1}, \dots , v_{n})$ 
if $v_{1}, \dots, v_{n}$ is positively oriented and letting 
$\Phi(x,v_{1}, \dots , v_{n}) = -\Psi_{x}(v_{1}, \dots , v_{n})$
if  not. This construction coincides with sending a top-order form to its
absolute value only if the form never changes sign. 

It is easy to
see that if we apply this procedure to the form $\Omega_{n-k}$ on the
oriented manifold $H_{n,n-k}^{+}$, we will obtain a top-order density
which descends to $\Phi_{n-k}$ on the space of unoriented $(n-k)$-flats.


 This section ends with one last construction relating the forms 
$\Omega_{n-k}$ to the powers of the symplectic form $\omega$.

Let $\Lambda$ be an oriented $(k+1)$-flat, let $H_{1}^{+}(\Lambda)$
be the set of oriented lines lying on $\Lambda$, and let 
$\Delta_{\Lambda}$ be the set of oriented $(n-k)$-flats which do not
intersect $\Lambda$ transversely. We define a fibration
$p : H_{n,n-k}^{+} \setminus \Delta_{\Lambda} \rightarrow H_{1}^{+}(\Lambda)$ 
by assigning to each $\zeta \in  H_{n,n-k}^{+} \setminus \Delta_{\Lambda}$
the line $\zeta \cap \Lambda$. The orientation of this line is 
determined as follows: if $\Pi \subset \Lambda$ is any oriented 
$k$-flat, then the line $\zeta \cap \Lambda$ is oriented so that
its intersection number with $\Pi$ in $\Lambda$ coincides with the
intersection number of $\zeta$ with $\Pi$ in $\R^n$. 

Since the fibration $p$ is oriented, it makes sense to compute 
$p_{*}\Omega_{n-k}$. The following result follows easily from the
last theorem.

\begin{prop} \label{reduction}
The form $p_{*}\Omega_{n-k}$ equals the pull back of the form 
$\omega^k$ to the set of all oriented lines contained in $\Lambda$. 
\end{prop}

 If we regard $\Lambda$ as a Finsler space with the Finsler metric
it inherits from the projective Finsler metric on $\R^n$, then 
$\Lambda$ is a projective Finsler space on its own right. The 
symplectic form on its space of geodesics, $H^{+}_{1}(\Lambda)$, 
coincides with the pull back of $\omega$ to $H^{+}_{1}(\Lambda)$
by its embedding into $H_{n,1}^{+}$. The form $\omega^k$ appearing
in the proposition is then nothing more than the Liouville form on the
space of geodesics on $\Lambda$.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Proof of the main theorem}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

We start by remarking that to prove the Crofton formula 
it suffices to show that if $D$ is a domain in a $k$-flat, then
\[
\mbox{\textit{vol}}(D) = 
{1\over \epsilon_{k}} \int_{\zeta \cap D \neq \emptyset} \Phi_{n-k} .
\]

 Because of the results in the previous section, it is more convenient
to work with the forms $\Omega_{n-k}$ than with the densities 
$\Phi_{n-k}$.  Note that if $D$ is a domain in an oriented $k$-flat
$\Pi$, then the integral of $\Phi_{n-k}$ over all $(n-k)$-flats 
intersecting $D$ equals the absolute value of the integral of 
$\Omega_{n-k}$ over all oriented $(n-k)$-flats which intersect 
$D$ and whose intersection number with $\Pi$ is positive. It 
suffices then to show that
\[
\mbox{\textit{vol}}(D) = 
{1\over \epsilon_{k}} \left| \int_{\zeta \cap D > 0} \Omega_{n-k} \right| .
\]
 
 Let us begin with the simplest case.

\begin{lemma}
If $D \subset \R^n$ is a domain in an oriented $(n-1)$-flat, then
\[
\mbox{\textit{vol}}(D) = {1 \over \epsilon_{n-1}}
\left| \int_{l \cap D > 0} \Omega_{1}\right| ,
\]
where $\epsilon_{n-1}$ is the volume of the Euclidean unit ball of dimension
$n-1$.
\end{lemma}

\begin{proof}
 By Theorem~\ref{general} the volume of $N$ equals
\[
{1 \over 2\epsilon_{n-1}} \int_{l \cap D > 0} |\omega^{n-1}|,
\] 
which in turn equals $\epsilon_{n-1}^{-1}$ times the absolute value of the
integral of $\omega^{n-1}$ over the set of all oriented lines which 
intersect $D$ positively. Since according to Theorem~\ref{powers-of-omega} 
$\Omega_{1} = \omega^{n-1}$, this finishes the proof.
\end{proof}

 The idea of the rest of the proof is to reduce everything to this 
simplest case by the following procedure:

Let $D$ be a domain in an oriented $k$-flat and  let $\Lambda$ be an oriented
$(k+1)$-flat containing it. Note that $\Lambda$ inherits a projective 
Finsler metric from that of $\R^n$ and that $D$ is a hypersurface in $\Lambda$.
Theorem~\ref{general} then tells us that
\[
\mbox{\textit{vol}}(D) = 
{1 \over \epsilon_{k}} \left| \int_{l \cap D > 0} \omega^{k} \right| .
\]

To go from an integral over oriented lines in $\Lambda$ to an
integral over oriented $(n-k)$-flats in $\R^n$ we make use of the
last construction of the previous section.

If $p : H_{n,n-k}^{+} \setminus \Delta_{\Lambda} \rightarrow H_{1}^{+}(\Lambda)$ is 
the fibration in Section 3, then by Proposition~\ref{reduction}
\[
\int_{l \cap D > 0} \omega^{k} = 
\int_{l \cap D > 0} p_{\ast}\Omega_{n-k} .
\]
Since the set of oriented lines in $\Lambda$ which intersect $D$ 
positively is exactly the image under $p$ of the set of all oriented 
$(n-k)$-flats which intersect $D$ positively, we have that
\[
\int_{l \cap D > 0} p_{\ast}\Omega_{n-k} =
\int_{\zeta \cap D > 0} \Omega_{n-k} .
\]
To summarize:
\[
\mbox{\textit{vol}}(D) = 
{1 \over \epsilon_{k}} \left| \int_{l \cap D > 0} \omega^{k} \right| =
{1 \over \epsilon_{k}} \left| \int_{l \cap D > 0} p_{*}\Omega_{n-k} \right| =
{1 \over \epsilon_{k}} \left| \int_{\zeta \cap D > 0} \Omega_{n-k} \right| .
\]
This finishes the proof of the Crofton formulas. \qed



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%% REFERENCES %%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

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