real intersection bodies. We use this result in Section \[CIB\], where we define complex intersection bodies and prove a Fourier characterization of intersection bodies: an origin symmetric complex star body $K$ in $\R^{2n}$ is a complex intersection body if and only if the function $\|\cdot\|_K^{-2}$ represents a positive definite distribution. We use this characterization in Section \[Char of CIB\] to show that the class of complex intersection bodies coincides with the class of real 2-intersection bodies in $\R^{2n}$ and, at the same time, with the class of generalized 2-intersection bodies, provided that bodies from the real classes possess symmetries determined by the complex structure of $\R^{2n}.$ The latter allows to extend to the complex case a result of Goodey and Weil [@GW] by showing that all symmetric complex intersection bodies can be obtained as limits in the radial metric of complex radial sums of ellipsoids. Finally, Section \[stab-hyper\] deals with stability in the complex Busemann-Petty problem for arbitrary measures and related hyperplane inequalities.

Complex intersection bodies of star bodies {#CIBofstarbodies}
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The theory of real convex bodies goes back to ancient times and continues to be a very active field now. The situation with complex convex bodies is different, as 