if there exists a finite Borel measure $\mu$ on $S^{n-1}$ so that for every even test function $\phi$ in $\R^n,$ $$\label{defkintgeneral}
\int_{\R^n} \|x\|_D^{-k} \phi(x)\ dx = 
\int_{S^{n-1}} \left(
\int_0^\infty t^{k-1} \hat\phi(t\xi)\ dt\right) d\mu(\xi).$$

The class of $k$-intersection bodies is related to a certain generalization of the Busemann-Petty problem in the same way as intersection bodies are related to the original problem (see [@K1] for details; this generalization offers a condition that allows to compare volumes of two bodies in arbitrary dimensions). An equivalent and probably more geometric way to define $k$-intersection bodies would be to say that these bodies are limits in the radial metric of $k$-intersection bodies of star bodies (see [@EMilman1] or [@Ru] for a proof of equivalence of this property to the original definition from [@K4]).

It was shown in [@K4] that an origin symmetric star body $K$ in $\R^n$ is a $k$-intersection body if and only if the function $\|\cdot\|_K^{-k}$ represents a positive definite distribution. By Theorem \[posdef\],

\[Char2\] An origin symmetric complex star body $K$ in $\R^{2n}$ is a complex intersection body if and only if it is a 2-intersection body in $\R^{2n}$ satisfying (\[rotation\]).

It was proved in [@KKZ Theorem 3] that every origin 