${\cal{R}}_c: C_c(S^{2n-1})\to C_c(S^{2n-1})$ defined by $${\cal{R}}_cf(\xi) = \int_{S^{2n-1}\cap H_\xi} f(x) dx.$$ Writing volume in polar coordinates, we get that for every complex star body $L$ in $\R^{2n}$ and every $\xi\in S^{2n-1},$ $$\label{v-r}
  |L\cap H_\xi| = \frac1{2n-2} \int_{S^{2n-1}\cap H_\xi} \|x\|_L^{-2n+2} dx 
  =  \frac1{2n-2} {\cal{R}}_c\left(\|\cdot\|_L^{-2n+2}\right)(\xi),$$ so the condition (\[intbody\]) reads as $$\label{int-radon}
  \|\xi\|_{I_c(L)}^{-2} =  \frac1{2\pi(n-1)} {\cal{R}}_c\left(\|\cdot\|_L^{-2n+2}\right)(\xi).$$ This means that a complex star body $K$ is a complex intersection body of a star body if and only if the function $\|\cdot\|_K^{-2}$ is the complex spherical Radon transform of a continuous positive $R_\theta$-invariant function on $S^{2n-1}.$ We use this observation in Section \[CIB\], where we introduce a more general class of complex intersection bodies (not depending on the underlying star body), like it was done in the real case in [@GLW]. But before that we need several facts connecting the Radon transform to the Fourier transform in the complex setting.

We use the techniques of the Fourier transform of distributions; see [@GS] for details. As usual, we denote by ${\cal{S}}(\R^n)$ the Schwartz space of rapidly decreasing infinitely differentiable functions (test functions) in $\R^n,$ and ${\cal{S}}^{'}(\R^n)$ is the space of distributions over ${\cal{S}}(\R^n).$

Suppose that $f$ is a locally integrable complex-valued function on $\R^n$ with 