theorem saying that an origin symmetric star body is uniquely determined by volume of its central hyperplane sections; see [@K1 Corollary 3.9].

If $K,L$ are origin symmetric complex star bodies in $\R^{2n},$ and their intersection bodies $I_c(K)$ and $I_c(L)$ coincide, then $K=L.$

The equality of intersection bodies means that, for every $\xi\in S^{2n-1},$ $|K\cap H_\xi|= |L\cap H_\xi|.$ By Theorem \[volume-ft\], homogeneous of degree -2 continuous on $\R^{2n}\setminus \{0\}$ functions $\left(\|\cdot\|_K^{-2n+2}\right)^\wedge$ and $\left(\|\cdot\|_L^{-2n+2}\right)^\wedge$ coincide on the sphere $S^{2n-1}$, so they are also equal as distributions on the whole of $\R^{2n}.$ The result follows from the uniqueness theorem for the Fourier transform of distributions.

Complex intersection bodies {#CIB}
===========================

We are going to define the class of complex intersection bodies by extending the equality (\[int-radon\]) to measures, as it was done in the real case in [@GLW]. We say that a finite Borel measure $\mu$ on $S^{2n-1}$ is $R_\theta$-invariant if for any continuous function $f$ on $S^{2n-1}$ and any $\theta\in [0,2\pi]$, $$\int_{S^{2n-1}} f(x) d\mu(x) = \int_{S^{2n-1}} f(R_\theta x) d\mu(x).$$ The complex spherical Radon transform of an $R_\theta$-invariant measure $\mu$ is defined as a functional ${\cal{R}}_c\mu$ on the space $C_c(S^{2n-1})$ acting by $$\left({\cal{R}}_c\mu, f \right) = \int_{S^{2n-1}} {\cal{R}}_cf(x) d\mu(x).$$

\[int\] An origin symmetric 