symmetric complex convex body $K$ in $\R^{2n}$ is a $(2n-4)$-, $(2n-3)$- and $(2n-2)$-intersection body in $\R^{2n}.$ It follows that

\[n&lt;4\]Every origin symmetric complex convex body in $\R^6$ and $\R^4$ is a complex intersection body.

This is no longer true in $\R^{2n},\ n\ge 4$ as shown in [@KKZ Theorem 4]. The unit balls of complex $\ell_q$-balls with $q>2$ are not $k$-intersection bodies for any $1\le k < 2n-4.$

Zhang in [@Zh3] introduced another generalization of intersection bodies. For $1\leq k\leq n-1$, the ($n-k$)-dimensional spherical Radon transform is an operator ${\cal{R}}_{n-k}:C(S^{n-1})\mapsto C(G(n,n-k))$ defined by $${\cal{R}}_{n-k}(f)(H)=\int_{S^{n-1}\cap H}f(x)dx,\quad H\in G(n,n-k).$$ Here $G(n,n-k)$ is the Grassmanian of $(n-k)$-dimensional subspaces of $\R^n.$ Denote the image of the operator ${\cal{R}}_{n-k}$ by X: $${\cal{R}}_{n-k}\left(C(S^{n-1})\right)=X\subset C(G(n,n-k)).$$ Let $M^+(X)$ be the space of linear positive continuous functionals on $X$, i.e. for every $\nu\in M^+(X)$ and non-negative function $f\in X$, we have $\nu(f)\geq0$.

(Zhang [@Zh3]) An origin symmetric star body $K$ in $\R^n$ is called a [*generalized $k$-intersection body*]{} if there exists a functional $\nu\in M^+(X)$ so that for every $f\in C(S^{n-1})$, $$\int_{S^{n-1}}\|x\|_K^{-k} f(x)d x= \nu({\cal{R}}_{n-k}{f}).$$

It is easy to see that every complex intersection body in $\R^{2n}$ is a generalized 2-intersection body in $\R^{2n}.$ If $K$ is a complex intersection 