is the intersection body of a star body if and only if the function $\|\cdot\|_K^{-1}$ is the spherical Radon transform of a continuous positive function on $S^{n-1}.$

A more general class of bodies was introduced in [@GLW]. A star body $K$ in $\R^n$ is called an [*intersection body*]{} if there exists a finite Borel measure $\mu$ on the sphere $S^{n-1}$ so that $\|\cdot\|_K^{-1}= R\mu$ as functionals on $C(S^{n-1}),$ i.e. for every continuous function $f$ on $S^{n-1},$ $$\label{defintbody}
\int_{S^{n-1}} \|x\|_K^{-1} f(x)\ dx = \left( {\cal{R}}\mu, f \right)= \int_{S^{n-1}} Rf(x)\ d\mu(x).$$

We introduce complex intersection bodies along the same lines. In Section \[CIBofstarbodies\] we define complex intersection bodies of complex star bodies, and in Section \[bus\] we study complex intersection bodies of convex bodies. While the complex version of Busemann’s theorem requires a serious effort, the extension of the Hensley-Borell theorem to the complex case follows from a result of Ball [@Ball1]. In Section \[radonandfourier\] we prove that the complex spherical Radon transform and the Fourier transform of distributions coincide (up to a constant) on a class of $(-2n+2)$-homogeneous functions on $\R^{2n}$ with symmetries determined by the complex structure. A similar result in the real case was crucial for the study of 