complex star body $K$ in $\R^{2n}$ is called a [complex intersection body]{} if there exists a finite Borel $R_\theta$-invariant measure $\mu$ on $S^{2n-1}$ so that $\|\cdot\|_K^{-2}$ and ${\cal{R}}_c\mu$ are equal as functionals on $C_c(S^{2n-1}),$ i.e. for any $f\in C_c(S^{2n-1}),$ $$\int_{S^{2n-1}} \|x||_K^{-2} f (x)\ dx = \int_{S^{2n-1}} {\cal{R}}_c f(\theta) d\mu(\theta).$$

Clearly, ${\cal{R}}_c\mu$ is a finite Borel $R_\theta$-invariant measure on $S^{2n-1}.$ Also, an easy consequence of self-duality of the complex spherical Radon transform (see Lemma \[self-dual\]) is that if $\mu$ has density $f$ on $S^{2n-1}$, then the measure ${\cal{R}}_c\mu$ has density ${\cal{R}}_cf.$ The latter, in conjunction with (\[int-radon\]), immediately implies that every complex intersection body of a star body is a complex intersection body in the sense of Definition \[int\].

Many results on real intersection bodies depend on the following Fourier characterization (see [@K5 Theorem 1]): an origin symmetric star body $K$ in $\R^n$ is an intersection body if and only if the function $\|\cdot\|_K^{-1}$ represents a positive definite distribution. Complex intersection bodies admit a similar characterization. To see the connection with the Fourier transform, combine the definition of the intersection body of a star body (\[def-intbody\]) with the result of Theorem \[volume-ft\]: for every $\xi\in S^{2n-1},$ $$\|\xi\|_{I_c(L)}^{-2} =  \frac1{\pi}|L\cap H_\xi| = 