[0,\infty),\ \theta\in S^{2n-1},$ $u\in {\cal{S}}(\R)$ is a non-negative test function on $\R,$ and $v$ is an arbitrary infinitely differentiable $R_\theta$-invariant function on $S^{2n-1}.$ Then $$\int_0^\infty r^{2n-3} \phi(r\theta) dr =  v(\theta) \int_0^\infty r^{2n-3} u(r) dr,$$ so the equality (\[eq3\]) turns into $$\int_{S^{2n-1}} \|\theta\|_K^{-2} v(\theta) d\theta 
=2\pi  \int_{S^{2n-1}} {\cal{R}}_c v(\xi)\ d\mu(\xi).$$ Since infinitely smooth functions are dense in $C_c(S^{2n-1}),$ the latter equality also holds for any function $v\in C_c(S^{2n-1}),$ which means that $K$ is a complex intersection body.

Characterizations of complex intersection bodies {#Char of CIB}
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Complex intersection bodies are related to two generalizations of the concept of a real intersection body. These relations allow to apply to the complex case many results established originally in the real case.

The concept of a $k$-intersection body was introduced in [@K3; @K4]. For an integer $k,\ 1\le k <n$ and star bodies $D,L$ in $\R^n,$ we say that $D$ is the $k$-intersection body of $L$ if for every $(n-k)$-dimensional subspace $H$ of $\R^n,$ $$|D\cap H^\bot|= |L\cap H|.$$ The class of $k$-intersection bodies was defined in [@K4] (see also [@K1 Section 4.2]) as follows.

Let $1\le k < n.$ We say that an origin symmetric star body $D$ in $\R^n$ is a $k$-intersection body 