the concept of an intersection body to the complex case.

Let $K$ and $L$ be origin symmetric star bodies in $\R^n.$ Following [@L], we say that $K$ is the [*intersection body of*]{} $L$ if the radius of $K$ in every direction is equal to the volume of the central hyperplane section of $L$ perpendicular to this direction, i.e. for every $\xi\in S^{n-1},$ $$\label{intbodyofstar}
\|\xi\|_K^{-1} = |L\cap \xi^\bot|,$$ where $\|x\|_K=min\{a\ge 0:\ x\in aK\}$, $\xi^\bot=\{x\in \R^n:\ (x,\xi)=0\},$ and $|\cdot|$ stands for volume. By a theorem of Busemann [@Bu1] the intersection body of an origin symmetric convex body is also convex. However, intersection bodies of convex bodies form just a small part of the class of intersection bodies. In particular, by results of Hensley [@H] and Borell [@Bor], intersection bodies of symmetric convex bodies are isomorphic to an ellipsoid, i.e. $d_{BM}(I(K), B_{2}^{n})\le c$ where $d_{BM}$ is the Banach-Mazur distance and $c>0$ is a universal constant.

The right-hand side of (\[intbodyofstar\]) can be written using the polar formula for volume: $$\|\xi\|_K^{-1}= \frac{1}{n-1} \int_{S^{n-1}\cap \xi^\bot} \|\theta\|_L^{-n+1} d\theta=
\frac{1}{n-1} {\cal{R}}(\|\cdot\|_L^{-n+1})(\xi),$$ where the operator ${\cal{R}}: C(S^{n-1})\to C(S^{n-1})$ is the spherical Radon transform defined by $${\cal{R}}f(\xi) = \int_{S^{n-1}\cap \xi^\bot} f(x) dx.$$ This means that a star body $K$ 