that any intersection body is the limit (in the radial metric topology) of finite radial sums of ellipsoids. This result has been extended by Grinberg and Zhang [@GZ] (see another proof in [@EMilman1]) to the case of generalized $k$-intersection bodies where the radial sum is replaced by the $k$-radial sum. Now we are going to prove a complex version of the result of Goodey and Weil. We do it by adjusting to the complex case the proofs from [@EMilman1] and [@KY Theorem 3.10].

We define the complex radial sum $K_{1}+^{c} K_{2}$ of two complex star bodies $K_{1}, K_{2}$ as the complex star body that has radial function $$\rho_{K_{1}+^{c} K_{2}}^{2}  = \rho_{K_{1}}^{2} + \rho_{K_{2}}^{2}  .$$ The latter can be written as $$\|\cdot\|_{K_{1}+^{c} K_{2}}^{-2} = \|\cdot\|_{K_1}^{-2} + \|\cdot\|_{K_1}^{-2}.$$

\[Thm:approximation\] Let $K$ be an origin symmetric complex star body in $\R^{2n}$. Then $K$ is a complex intersection body if and only if $ \|\cdot\|_K^{-2}$ is the limit (in the metric of the space $C_c(S^{2n-1})$) of finite sums $\|\cdot\|_{E_1}^{-2}+\cdots+ \|\cdot\|_{E_m}^{-2} $, where $E_1$,...,$E_m$ are complex ellipsoids in $\R^{2n}$ (i.e. those ellipsoids in $\R^{2n}$ that are complex convex bodies).

In other words, an origin symmetric complex star body is an intersection body if and only 