body, then there exists an even $R_\theta$-invariant measure $\mu$ on $S^{2n-1}$ such that for every $f\in C(S^{2n-1}),$ $$\int_{S^{2n-1}} \|x\|_K^{-2} f(x) dx = \int_{S^{2n-1}} {\cal{R}}_cf(\xi) d\mu(\xi)$$ $$=\int_{G(2n,2n-2)} {\cal{R}}_{2n-2} f(H) d\nu(H),$$ where $\nu$ is a measure on $G(2n,2n-2)$ which is the image of $\mu$ under the mapping $\xi\mapsto H_\xi$ from $S^{2n-1}$ to $G(2n,2n-2).$ The measure $\nu$ can be considered as a positive continuous functional on $X$ acting by $$\nu({\cal{R}}_{n-k}f)= \int_{G(2n,2n-2)} {\cal{R}}_{n-k}f(H) d\nu(H),$$ which implies that $K$ is a generalized 2-intersection body in $\R^{2n}.$

On the other hand, it was shown in [@K4] (see also [@K1 Theorem 4.23]) that every generalized $k$-intersection body in $\R^n$ is a $k$-intersection body. So we have shown the following

An origin symmetric complex star body $K$ in $\R^{2n}$ is a complex intersection body if and only if it is a generalized $2$-intersection body in $\R^{2n}$ satisfying (\[rotation\]).

Let us point out that the latter result is surprising. Combined with Corollary [\[Char2\]]{} implies that under the invariance assumption (\[rotation\]) the class of $2$-intersection bodies is exactly the class of generalized $2$-intersection bodies. Without the invariance assumption (\[rotation\]) this is no longer true as it follows from an example of E. Milman [[@EMilman2]]{}.

Goodey and Weil in [[@GW]]{} proved 