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\topmatter
\title
\nofrills{
%                {\bf To appear in: Acta Sci.\ Math.} \\ \\
\bigrm  
	       The set of\,\,invertible \\ \\ Radon transforms
}
\endtitle
\author
Tibor \'Odor 
\endauthor
\rightheadtext\nofrills{\sevenrm 
                 The set of invertible Radon transforms
}
\leftheadtext\nofrills{\sevenrm 
                              Tibor \'Odor
}
\address
Math.\ Inst.\ of the Hung.\ Acad.\ of Sci., 
P.O.B.\ 127, H-1364 Hungary
\endaddress
\email
odor\@math-inst.hu
\endemail
\thanks
Partially supported by Hung.\ Nat.\ Found.\ for Sci.\ Research, number 4427
and the Deutsche Forschungsgemeinschaft.
\endthanks
\subjclass
0052, 44A05, 53C65
\endsubjclass
\keywords
convex body, Radon-transform, $G_\delta$-set, invertibility, kernel, 
smoothness
\endkeywords
\abstract
 A general class of Radon transforms on the sphere is defined using convex
bodies. We show that the invertible Radon-transformations in this class
constitute a $G_{\delta}$ set.
\endabstract
\endtopmatter
\document
\vskip .4cm
\heading 
                            1.\ Introduction
\endheading
\vskip .2cm
 The invertibility of generalized Radon transforms was investigated by
several authors [Guillemin--Sternberg, 1977], [Mukhometov, 1977],
[Mukhometov, 1981], [Boman, 1984], [Boman, 1989]. Most of the results shows
that among fairly general conditions Radon transforms are invertible, or at
least admit finite dimensional kernel on a compact set. One would expect
that every ``nice'' Radon transform has this property. But the paper of L.\
Zalcman [Zalcman, 1980] refers to lot of results not fitting this frame. As
one can observe, the reson of weird behavior that general transforms on
compact manifolds are not Fredholm operators.

 We investigate these type of transforms, when and our work was inspired by
R.\ Schneider's works on the integration of $L^2$ functions on the sphere
[Schneider, 1969] and independent studies of general Radon transforms on the
spehere motivated by the Ulam floating body problem [\'Odor, 1999].

  Let  $\<\,\, ,\, \,\>$ be  the inner product in the $d$-dimensional
Euclidean space $\Bbb E^d$,  $ S^{d-1}$ is the unit sphere in $\Bbb E^d$, 
and $dx$ is the normalized surface measure on the sphere $S^{d-1}$.

 R.\ Schneider [Shneider, 1969], using the Funk-Hecke theorem and elementary
properties of Gegenbauer polynomials, proved that the transform
$$
     (R_{\alpha}f)(\omega)=\int_{\<\omega,x\>=\cos\alpha} f(x)\,dx
$$
 for functions $f\in L^2( S^{d-1})$ is not invertible for a countable
dense set of $\alpha\in (0,\pi/2)$, where $d\ge 3$. In other words
the invertible Radon transforms constitute a $G_{\delta}$-set among
the set of the transforms $R_{\alpha}$ on the interval
$(\alpha\in(0,\pi/2))$. The kernels of the transformations $R_\alpha$ 
can be determined explicitly by the help of  spherical harmonics.

 If $d=4$ then easy manipulation with Gegenyields that the kernel for
radiuses $r$ for which $\frac{\arccos1/r}{\pi}$ is rational is infinite
dimensional. (This immediately shows, that $R_\alpha$ is not a Fredholm
operator, and its backprojection is not an elliptic pseudo differential
operator.) 

 For other dimensions, except $d=1$ and $d=2$ it is not known whether the
kernel of $R_\alpha$ can be infinite dimensional or not for
$\alpha\ne\pi/2$. The author conjectures, that it cannot. This conjecture is
supported by heavy experimenting by {\bf Maple} by computing resultants of
the appropriate

\define\clp{\operatorname{cl_p^*\,}}
\define\Lp{L^{p*}(M)}
\define\Rpc{R_K^{p*}}
\redefine\Rp{R_K^{p}}
\define\Fsig{F_\sigma}
\define\Kd{\Cal K^d(M)}
\define\Imm{\operatorname{Im\,}}
\define\Gdel{G_\delta}
\define\Lpp{L^{p}(M)}

 We investigate the problem of invertibility in a more general framework,
and we get the appropriate result in Section 2.  For every convex body $K$
containing a fixed strictly convex body $M$ in its interior, we define caps
on $\partial M$ using boundary points of $K$.

 If $k$ is a boundary point of $K$ then the cap belongs to $k$ is the set of
those points of $\partial M$ which are visible from the point $k$.  Assuming
that $M$ contains the origin $O$, project these caps to the unit sphere
$S^{d-1}$ from $O$. Integrating $L^p$ for $(1<p\le\infty)$ functions on
these sets we define a class of generalized Radon transforms on the unit
sphere $S^{d-1}$.

 We prove that the set of invertible Radon transforms on $L^p$ for
$(1<p\le\infty)$ constitute a $G_\delta$ set with respect to the Hausdorff
metric of the set of convex bodies containing $M$ in their interior.

% For the sake of simplicity,
% we have not introduced strange and complicated definitions and
% notations to generalize to the greatest possible extent
% instead, we have tried to keep the paper readable by
% restricting ourself to a smaller class. Roughly speaking,
% we believe that our method works  with some minor modification
% on almost any ``natural'' set of Radon transforms
% with natural topology. On the other hand one must be careful with
% the function spaces. Our method works only
% on normed dual spaces. We use very strongly the weak*--compactness
% of the unit ball in the function spaces investigated in this article.

 It would be also interesting to investigate the invertibility of Radon
transforms on the space of continuous functions, integrable $(L^1)$
functions and the infinitely differentiable functions on the sphere
$S^{d-1}$. But for these spaces our methods does not work. On the other
hand, the choice of convex bodies is somehow arbitrary, and we choosed them
only to have an easy language for defining the Radon transforms.

% In the last Section we investigate the invertibility  for
% pairs of Radon  transforms, and consider the
% smoothness of the kernel. We always obtain ``nice'' sets. 

\vskip .2cm
\heading 
                  2. The set of invertible transforms
\endheading
\vskip .2cm
\redefine\M{\partial M}
\redefine\S{S^{d-1}}

 A convex body is a convex, compact set with non-void interior in the
Euclidean space $\Bbb R^d$. Let $M$ be a fixed strictly convex body
containing the unit sphere $S^{d-1}$ in its interior.

 Let $\Cal K^d(M)$ be the set of convex bodies in $\Bbb R^d$ containing the
body $M$ in their interior and endow $\Cal K^d$ with the Hausdorff distance
$d_H$. It is well known that $\Cal K^d$ is a localcompact metric space. Let
$dx$ be the normalized surface measure on the boundary $\partial M$ of the
body $M$. Project the surface $\M$ to $\S$ from the origin by the projection
$\pi$. Let $dm$ be the projected surface measure.

 Let $L^p(\S)$ denote the spaces of the $L^p$ functions on $\M$. Using the
previously defined projection $\pi$ we can identify the function spaces
$L^p(S^{d-1})$ and $L^p(\partial M)$ and endow them with the equivalent
norms $|\,.\,|_p$.

 The boundary points of $M$ visible from the point $x\in\Bbb R^d\setminus M$
is $M(x)=\{m\in\M:\,\overline{mx}\cap\interior M=\emptyset\}$.

 Let $1<p\le\infty$ be a real number and let $1\le q<\infty$ its dual pair,
i.e.\ for which $1/p+1/q=1$ holds. Let $k\in\partial K$ and $f\in L^p(M)$.

We define the Radon transform $R_K ^p:L^p(\S)\to L^q(\S)$ on the
sphere $S^{d-1}$  by the formula. $$(R_K^p f)(k/|k|)=\int_{\pi M(k)}
f(m)\,dm $$ As $\int_{\pi M(k)} f(m)\,dm\le\int_{\S} |f(m)|\,dm=|f|_1
\le|f|_p$, the transform $R_K^p$ maps the space $L^p (\S)$ to the
space $L^q(\S)$.

\proclaim{ Theorem 1}
  The set $$I_M^p=\{\,K\in\Cal K^d(M): R_K^p
\text{\,\,is invertible on $L^p(\S)$}\}$$
 is a $G_\delta$ set in $\Cal K^d(M)$ for any $1<p\le\infty$.
\endproclaim
 Before proving the theorem, we have to show the following crucial lemma. It
contains the main tools and ideas of this paper.

 If $X$ is a Banach space let $X^*$ be its dual space. Let $cl^{*}_p$ be the
weak*-closure operation on $L^{p*}(\S)$. Let $\Imm$ and $\Imm_+$ be the
images of $L^p(\S)$ and the non negative functions on $L^p(\S)$,
respectively. If $A:L^p(\S)\to L^q(\S)$ is a linear operator,
$A^*:L^{p*}(\S)\to L^{q*}(\S)$ is the adjoint of $A$. Let $\Ker A$ denote
the kernel of the linear operator $A$.

 For any $1\le p<\infty$, $L^{p*}=L^{q}$, where $1/p+1/q=1$ and
$A^*:L^p(\S)\to L^q (\S)$ [Rudin, 1973]. 

 Let $I_M^p(f)=\{\,K\in\Cal K^d(M):\, f \notin \clp \Imm\, R_K^{p*}\}$ be
the set of those convex bodies $\Cal K^d(M)$, for which a given function
$f\in L^{p*}(\S)$ is not contained in the weak*-closure of the image of the
transformation $R_K^{p*}$.

\proclaim{ Lemma 1}
 $I_M^p(f)$ is an $F_\sigma$ set in $\Cal K^d(M)$ 
 for any $f\in L^{p*}(\S)$ and $1<p\le\infty$.
\endproclaim

\demo{Proof}
Let 
$$
S(f)=\{\,\mu\in L^p(\S):\,\int\mu(m)\cdot f(m)\,dm=1\},
$$
and let 
$$
		 S_r(f)= \{\,\mu\in S(f): \,|\mu|_p\le r\}.
$$

 We know, that there exists a positive $r_*>0$, for which
$S_r(f)\ne\emptyset$, if $r>r_*$. Define the function 
$\nu_r:\Cal K^d(M)\to\Bbb R$ for every fixed $r\ge r_*$ as follows:
$$  
	      \nu_r(K,\mu) =\int_{\S}(R_K^p\mu)^2(m)\,dm\qquad
			      \text{and}\qquad
	     \nu_r(K) =\inf\limits_{\mu\in S_r(f)}\nu_r(K,\mu).
$$

 We can easily check that the transformation $\nu_r(K,\mu)$ is
weak*-continuous on $S_r(f)$ for every fix $r\ge r_*$. We know that $S_r(f)$
is weak*-compact if $r\ge r_*$ [Rudin, 1973]. So
$$
\nu_r(K)=\min\limits_{\mu\in S_r(f)}\nu_r(K,\mu).
$$ 

 That is, $\nu_r(K)=\emptyset$ is equivalent to 
$\Ker R_K^{p*}\cap S_r(f)\ne\emptyset$. We can easily check that 
$\nu_r:\Cal K^d(m)\to\Bbb R$ is a lower-semi-continous functional on 
$\Cal K^d(M)$, for every fix $r\ge r_*$. Because of this fact, the set
$$
\Cal Z_r=\{\,K\in\Cal K^d(M):\,\nu_r(K)\le0\}= \{\,K\in\Cal K^d(M):\,\nu_r(K)=0\}
$$
 is a closed set in the space of convex bodies $\Cal K^d(M)$ containing $M$
in their interior. It is well known that $f\in \clp\Imm R_K^{p*}$\, if and
only if \,$\int f(m)\cdot\mu(m)\,dm=0$ for every $\mu\in\Ker R_K^{p*}$
[Rudin,1973]. It is clear that the following two statements are equivalent:

1.\ There exists $\mu\in\Ker\,R_K^p$ such that 
 $\int\mu(m)\cdot f(m)\,dm\ne0$

2.\ There exists $r_*>0$ such  that if 
 $r>r_*$, then $\nu_r(K)=0$.\newline
(It comes from the fact that $S_r(f)\cap \Ker\,R_K^p\ne\emptyset$ if $r$ is
big enough.) So the set 
$$
		    \Cal Z=\bigcup_{r>0}\Cal Z_r=\{\,K:\,
		\text{$\exists r>0$ such that $\nu_r(K)=0$}\}
		    =\{\,K:\,f\notin \clp\Imm R_K^{p*}\}
$$
 is an $F_\sigma$ set. That is, the complement of $\Cal Z$ (which is
$I_M^p(f)$) is a $G_\delta$ set in the locally compact metric space 
$\Cal K^d(M)$.
\qed \enddemo
 After   Lemma 1,  we can easily prove Theorem 1:
\demo{Proof of Theorem 1}
 Let $h_0,h_1,\dots,h_n\dots,\,\,n\in\Bbb N$ be a weak*-dense set in 
\newline $L^{p*}(\S)$. This type of set  always
exists [Rudin, 1973, 105 pp. Exercise 1.a]. Let
$$
I_M^p=\bigcap_{i\in\Bbb N}I_M^p(h_i)
=\{\,K:\,\clp\Lin(h_i)_{i\in\Bbb N}\subset\,\clp\Imm R_K^{p*}\}=
$$
$$
		       \{\,K:\clp\Imm R_K^*=L^q(M)\}.
$$
 But $\Ker R_K^p={}^\perp(\Imm R_K^{p*})={}^\perp(\clp\Imm R_K^{p*})={}^\perp(L^q(\S))=\{0\}$,
if $K\in I_M$ [Rudin, 1973]. 
So $$I_M=\{\,K:\,R_K^p\text{\  is an invertible transformation}\}.$$
We know from Lemma 1
that $I_M^p(h_i)$ is a $G_\delta$ set for every $i\in\Bbb N$.
So $I_M^p=\bigcap_{i\in\Bbb N}I_M^p(h_i)$ is a $G_\delta$ set.
\qed
\enddemo
We can prove similarly that if $H\le L^q(\S)$ is a subspace, 
then $$I_M^p(H)=\{\,K\in\Cal K^d(M):\,H\le\clp
\Imm R_K^{p*}\}\quad\text{and}\quad$$ $$ K_M^p(H)=
\{\,K\in\Cal K^d(M):\,H^\perp\le\Ker R_K^p\}$$
is a $G_\delta$ set in $\Cal K^d(M)$.
It is interesting to investigate the  set
$$\Cal I_M^p(f)=\{\,K\in\Cal K^d(m):\,f\in\Imm R_K^p\}.$$

\proclaim{Lemma 2}
$\Cal I_M^p(f)$ is an $F_\sigma$ set in $\Cal K^d(M)$, where
$f\in L^{p*}(M)$ is a fixed function,
 $M$ is  a fixed  strictly convex body
 and $1<p\le\infty$.
\endproclaim
\demo{Proof}
Let $B_r^p=\{\,\mu\in L^p(M): |\mu|_p\le r\}$
be a closed ball with radius $r>0$.
We  define the functional $\nu_r:\Cal K^d(M)\to \Bbb R$
on the following way:
$$\nu_r(K)=\inf\limits_{\mu\in B_r^p}\int\limits_{\S}
(R_K^p\mu-f)^2(m)\,dm.$$
If $K$ and $r$ fixed, the functional $\nu(K,\mu)=\int_{\S}
(R_K^p\mu-f)^2(m)\,dm$ is 
weak*-continuous on $B_r^p$ because of the Banach--Alaoglu theorem
[Rudin, 1973].
So
 $$\nu_r(K)=\min\limits_{\mu\in B_r^p}\int\limits
_{\S}(R_K^p\mu-f)^2(m)\,dm.$$
$(*)$ So $(f\in\Imm R_K^p)$ is equivalent to the
existence of $r_0$, 
such that if $r\ge r_0$, then
$\nu_r(K)=0$.

It is clear that $\nu_R$ is a lower semi continuous functional.

Because of the lower-semi-continuity and the non-negativity of $\nu_r$,
$$\Cal Z_r'=\{\,K:\,\nu_r'(K)\le0\}=\{\,K:\,\nu_r(K)=0\}$$
is a closed set in $\Cal K^d(M)$. So
$$\align
\Cal Z&=\bigcup_{r>0}\Cal Z_r'=\{\,K\in\Cal K^d(M): \exists r\,_0
\text{that if $r\ge r_0$
then $\nu_r'(K)=0$}\}\\
&=\{\,K\in\Cal K^d(M): \exists\,\mu_K\in L^p(M),\,\,f=R_K^p\,\mu_K\}=
\{\,K\in\Cal K^d(M):f\in \Imm R_K^p\}
\endalign$$
is an $F_\sigma$ set, that is $\Cal I_M^p(f)$ is a $G_\delta$ set.
\qed \enddemo

In this case, we can not show directly, as in the Proof of Theorem  1 that
$\Cal I_M^q(H)=\{\,K\in\Cal K^d(M): H\le\Imm R_K^p\}$
(where $H\le L^q(\S)$ is a closed subspace) is an $F_\sigma$ set. We 
can only show 
that this set is an $F_{\sigma\delta}$ set. So we pose the following 
question:
\proclaim{Question:} Let $H\le L^{p*}(\S)$ be a closed subspace.
Is it true or not
 that $\Cal I_M^p(H)$ is an $F_\sigma$ set?
\endproclaim
\proclaim{Lemma 3} Let $H\le L^{p*}(\S)$ be a linear subspace, and 
assume that $H$ has
a countable linear base, that is $H$ is a linear hull
of countbly many functions
$h_i\in L^{p*}(\S)$, where $i\in \Bbb N$. 
In this case $\Cal I_M^q(H)$ is an $F_\sigma$ set.
\endproclaim
\demo{Proof}
Let  $\Cal L^p(\S)=
\prod\limits_{i\in\Bbb N}
L^p(\S)$ be the product of countably many
$L^p(\S)$ spaces and endow this space with the product topology of the 
weak*-topologies.

Let $\Cal B_r^p=\prod\limits_{i\in\Bbb N}B_r^p$ be 
the products of the closed balls with radius $r>0$.



Let 
$$ \nu_r(K)=\inf\limits_{(\mu_i)_{i\in\Bbb N}\in\Cal B_r^p}
\sum\limits_{i\in\Bbb N}\,\,
2^{-i}\cdot\int\limits_{\S}(R_K^p\mu_i-h_i)^2(m)\,dm.$$

From this point everything is similar to the proof of the Lemma 2
with some modification in
the point $(*)$, as we are working with the product topology.
\qed \enddemo
\vskip .4cm
\heading 2. The kernels of  Radon transforms
\endheading
\vskip .4cm

Let $\Cal K_n^p(M)=\{\,K\in
\Cal K^d(M):\,\dim \Ker R_K^p=n\}$ be the set of those convex bodies 
for which the Radon transform $R_K^p$ has 
exactly $n$-dimensional kernel on the function space $L^p(\S)$,
where $n\in\Bbb N^*=\Bbb N\cup\{\infty\}$, and $1<p\le\infty$.

\proclaim{Lemma 4}
$\Cal K_n^p(M)$ is a a $G_{\delta\sigma}$
 set, if $n\in\Bbb N$ and  $1<p\le\infty$.
\endproclaim
\demo{Proof}
Let $\zeta_n^p(M)=\{\,K\in\Cal K^d(M):\,\dim \Ker R_K^p\ge n\}$
be the set of those convex bodies for which the 
 Radon transform $R_K^p$ has 
at least  $n$-dimensional kernel on the function space $L^p(\S)$.

 Let $(h_i)_{i\in\Bbb N}$ be a countable, linearly independent weak${}^*$
dense set in $L^{p*}(\S)$, that is $\clp\Lin(h_i) _{i\in\Bbb N}= L^{p*}(M)$
[Rudin, 1973, pp. 105, Exercise 1.a]. In this case using the Banach--Alaoglu
theorem
$$
\zeta_n^p(M)=\bigcup_{i_1<i_2<\dots<i_n} \{\,K\in\Cal K^d(M):\,
h_{i_j}\notin\clp\Imm R_K^{p*},\,\,\,(1\le j\le n)\}
$$
 is an $F_\sigma$ set, because of  Lemma 1 and the statement (1.1). 

So $\Cal K_M^d(M)=\zeta_n^p(M)\setminus\zeta_{n+1}^p(M)$. This set is an
$F_\sigma\setminus F_\sigma$ set,
that is a $G_{\delta\sigma}$ set.
\qed \enddemo

\proclaim{Corollary}
 The set $\Cal K_\infty^p(M)$ is $G_{\delta\sigma}$ if $1<p\le\infty$.
\endproclaim
\demo{Proof}
We know that $\Cal K_\infty^p(M)=(I_M^p)^c\setminus\bigcup_{n\ge1}
\Cal K_n^p(M)$.
 Theorem 1 states that $(I_M^p)^c$ is an $F_\sigma$  set.  Lemma 4
states that $\bigcup_{n\ge1}\Cal K_n^p(M)$ is an $F_\sigma$ set. So 
$\Cal K_\infty^p(M)$ is
an $F_\sigma-F_\sigma$, that is a 
$G_{\delta\sigma}$ set.
\qed \enddemo
We give an another proof of the  Theorem 1.
\demo{Second Proof of  Theorem 1}
Let $(h_i)_{i\in\Bbb N}$ be a linearly independent set in $\Lp$ and assume
that $\text{cl}_p\,\Lin(h_i)_{i\in\Bbb N}=\Lp$.

Let $S_i=\{\mu\in L^p(\S):\int\mu(m)\cdot h_i(m)\,dm=1\}$. Furthermore let
$B^p=\{\mu\in L^p(\S): |\mu|_p\le1\}$ and
$$
\nu_i(K,\mu)=\int_{\S}(\Rp\mu)^2(m)\,dm\qquad \text{and}\qquad
\nu_i(K)=\inf\limits_{\mu\in B^p\cap S_i}\nu(K,\mu).
$$

It is easy to see that:\newline

1. $\nu_i(K,\mu)$ is a weak*-continuous function on $B^p$\newline

2. $B^p\cap S_i$ is a weak*-compact set\newline

3. $\nu_i:\Kd\to\Bbb R$ lower-semi-continous, so
$\nu_i(K)=\min\limits_{\mu\in B^p\cap S_i}\nu(K,\mu).$\newline
We can easily check that the following two statements are equivalent:\newline

1. There exists non zero $f\in\Ker\Rp\subset L^p(\S)$\newline

2. There exists $i\in\Bbb N$ for which  $\nu_i(K)=0$.\newline
So $\bigcup\limits_{i\in\Bbb N}\{K\in\Kd:
\nu_i(K)=0\}$ is an $F_\sigma$ set, with its complement 
$I^p_K$ is  $G_\delta$.
\qed \enddemo

 Using backprojection which yield an elliptic pseudo differential operator
and applying their developed theory, for sufficiently smooth ($C^r$ for
$r\ge d$) and a fairly general class of Radon transforms (which satisfy the
so called Bolker assumption) we can prove that the kernel is $C^{r-d}$ and
finite dimensional on a compact set [Guillemin-Sternberg, 1977]. As we saw,
this does not apply in our case. This inspired the following studies.

Let $1<p<s\le\infty$ be fixed real numbers. Let 
$\Cal S^{p,s}(M)=\{K\in\Kd:\Ker\Rp\subset L^s(M)\}$ be the set of those 
convex bodies for which the $L^p$-kernel of the transformation $R^p_K$
coincides with the $L^s$-kernel of the transformation $R^s_K$. We can prove
the following theorem:
\proclaim{Theorem 2}
$\Cal S^{p,\infty}(M)$ is a $G_{\delta\sigma}$ set in $\Kd$, where
$1<p<\infty$.
\endproclaim
\demo{Proof} Let 
$\Cal S_0^{p,s}(M)=\{K\in\Cal S^{p,s}(m):1\le \dim\Ker\Rp<\infty\}$ be the
set of those convex bodies from $S^{p,s}(M)$ for which the kernel of $R^p_K$
is non-trivial but finite dimensional.

 Let $\Cal S_\infty^{p,s}(M)= \{K\in\Cal S^{p,s}(m):t \dim\Ker\Rp=\infty\}$
be the set of those convex bodies from $S^{p,s}(M)$ for which the kernel of
$R^p_K$ is infinite dimensional.


\proclaim{Lemma 6} $\Cal S_0^{p,s}(M)$is an $\Fsig$ set in $\Kd$.
\endproclaim
\demo{Proof} Let $n\ge1$ be an integer. In this case let
$$\Cal H_n^{p,s}=\zeta_{n}^s(M)\setminus\zeta_{n+1}^p(M)=
\Cal K_n^s(M)\setminus\zeta_{n+1}^p(M)=
\{K:\dim\Ker \Rp=n,\,\,\Ker \Rp\subset L^s(M)\}.$$
Clearly, $\Cal H_n^{p,s}$ is a $G_{\delta\sigma}$
 set. So $\Cal S_0^{p,s}(M)=
\bigcup_{n\ge1}\Cal H_n^{p,s}$ is an $G_{\delta\sigma}$ set too.
\qed \enddemo

\proclaim{Lemma 7} $\Cal S_\infty^{p,\infty}(M)=\emptyset.$
\endproclaim
The statement of this lemma says in other words that if the 
$L^p$-kernel of
a Radon transformation is constituted by bounded functions,
where $1<p<\infty$, then this kernel
is finite dimensional.

\demo{Proof} The proof of this lemma is an easy consequence of 
Groth\-en\-di\-eck's following theorem:\newline

Suppose that $0<p<\infty$ and

(a.) $m$ is a probability measure on $\S$ \newline

(b.) $S$ is a closed subspace of $L^p(\S)$\newline

(c.) $S\subset L^\infty(\S)$.\newline

Then $S$ is finite dimensional.
(For the proof see: [Rudin, 1973].)\newline
\define\clpp{\operatorname{cl_p}} 
Assume that  $K\in\Cal S_\infty^{p,\infty}(M)$.
If $(\Ker R_K^\infty)\subset L^\infty(\S)$ is not a closed subspace of
 $\L^p(\S)$, then $(\Ker R_K^\infty)\ne\clpp\Ker R_K^\infty
 \subset\Ker\Rp$,
where $\clpp A$ denotes the $p$-norm closure of the set $A\subset L^p$. So
$(\Ker R_K^p)\setminus(\Ker R_K^\infty)\ne\emptyset$, which contradicts
the assumption that $K\in\Cal S_\infty^{p,\infty}(M)$, because the set
 $(\Ker R_K^p)$ is a $p$-norm closed set.

So we can assume that $(\Ker R_K^\infty)\subset L^\infty(\S)$
 is a closed subspace
of $L^p(\S)$. That is because of the theorem of Grothendieck
$\dim\Ker R_K^p=n<\infty$.
This contradicts to the assumption $K\in\Cal S_\infty^{p,\infty}(M)$.
So $\Cal S_\infty^{p,\infty}(M)=\emptyset$.
\qed \enddemo
The proof of our theorem is the consequence of the previous two lemmas.
\qed \enddemo
Because of the fact that in the case $1<p<s<\infty$ there exists 
an infinite dimensional subspace
of $L^p$ which is $L^s$ closed, this proof does not work in general. 

We state
here the following conjecture about these cases:
\proclaim{Conjecture}
If $1<p<s\le\infty$, then the sets $\Cal S^{p,s}(M)$ are $\Fsig$ sets in $\Kd$.
\endproclaim

\vskip .2cm
\heading 
                           Acknowledgements
\endheading
\vskip .2cm

I would like to thank to the unknown referees the valuable comments and
information which helped to improve substantially the presentation
of the paper.

\vskip .4cm
\Refs
\widestnumber\key{HKM87aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa}

\ref\key Boman, 1984
\by J. Boman
\paper Uniqueness theorems for generalized Radon transforms
\jour Constructive Theory of Functions'84, Sofia
\yr 1984
\pages 173--176
\endref \vskip .1cm

\ref\key Guillemin, 1985
\by V. Guillemin
\paper On some results of Gelfand in integral geometry
\book  in Proc. Sympos. Pure. Math.
\vol 43
\yr 1985
\pages 149--155
\endref \vskip .1cm

\ref\key Guillemin--Sternberg, 1977
\by V. Guillemin--S. Sternberg
\book Geometric asymptotics
\publ Amer Math. Soc.
\publaddr Providence, Rhode Island
\yr 1977
\endref \vskip .1cm

\ref\key Mukhometov, 1977
\by R. G. Mukhometov
\paper The problem of recovery of a two-dimensional Riemann
metric and integral geometry (in Russian)
\jour Dokl. Akad. Nauk SSSR
\vol 22
\yr 1977
\pages 32--35
\moreref
\paper (English translation in)
\jour Soviet Math. Dokl.
\vol 18
\yr 1977
\pages 27--31
\endref \vskip .1cm

\ref\key Mukhometov, 1981
\by R. G. Mukhometov
\paper A problem of reconstructing a Riemannian metric (in Russian)
\jour Sibirsk. Mat. Zh.
\vol 22
\yr 1981
\pages 119--135
\moreref
\paper (English translation in)
\jour Siberian Math. J.
\vol 22
\yr 1981
\pages 420--433
\endref \vskip .1cm

\ref\key \'Odor, 1999
\by T.\ \'Odor
\paper Ulams floating body problem, 
\jour {\sl (submitted)}
\endref \vskip .1cm

\ref\key Rudin, 1973
\by W. Rudin
\book Functional analysis
\publ McGraw--Hill Book Company
\publaddr New York
\yr 1973
\endref \vskip .1cm

\ref\key Schneider, 1969
\by R. Schneider
\paper Functions on a sphere with vanishing integral over certain
subspheres
\jour J. of Math. Anal. and Appl.
\vol 26
\yr 1969
\pages 381--384
\endref \vskip .1cm

\ref\key Zalcman, 1980
\by L.\ Zalcman
\paper Offbeat integral geometry
\jour Amer.\ Math.\ Monthly
\vol 87
\yr 1980
\pages 161--175
\endref \vskip .1cm
\endRefs
\enddocument
