We introduce complex intersection bodies of complex star bodies using a definition under which these bodies play the same role in complex convexity, as their real counterparts in the real case. We use the notation $|K|$ for the volume of $K;$ the dimension where we consider volume is clear in every particular case.

\[intof\] Let $K, L$ be origin symmetric complex star bodies in $\R^{2n}.$ We say that $K$ is the complex intersection body of $L$ and write $K=I_c(L)$ if for every $\xi \in \R^{2n}$ $$\label{def-intbody}
 |K\cap H_\xi^{\perp}|= |L\cap H_\xi|.$$

Since $K\cap H_\xi^{\perp}$ is the two-dimensional Euclidean circle with radius $\|\xi\|_K^{-1}$, (\[def-intbody\]) can be written as $$\label{intbody}
 \pi  \|\xi\|_{I_c(L)}^{-2} =  |L\cap H_\xi|.$$ All the bodies $K$ that appear as complex intersection bodies of different complex star bodies form [*the class of complex intersection bodies of star bodies*]{}. In Section \[CIB\], we will introduce a more general class of [*complex intersection bodies*]{}.

The Radon and Fourier transforms of $R_\theta$-invariant functions {#radonandfourier}
==================================================================

Denote by $C_c(S^{2n-1})$ the space of $R_\theta$-invariant continuous functions, i.e. continuous real-valued functions $f$ on the unit sphere $S^{2n-1}$ in $\R^{2n}$ satisfying $f(\xi)=f(R_\theta(\xi))$ for all $\xi\in S^{2n-1}$ and all $\theta\in [0,2\pi].$ The complex spherical Radon transform is an operator 