at exactly two points and the Minkowski functional of $K$ defined by $$\|x\|_K= \min\{a\ge 0:\ x\in aK\},\qquad \forall x\in \R^n$$ is a continuous function on $\R^n.$ The radial function of $K$ is given by $$\rho_K(x) = \max\{a>0:\ ax\in K\}.$$ If $x\in S^{n-1},$ then $\rho_K(x)$ is the radius of $K$ in the direction of $x.$ Note that for any unit vector $\xi$, $\rho_K(\xi)=\|\xi\|_K^{-1}.$ The radial metric in the class of star bodies is defined by $$\rho(K,L)=\max_{\xi\in S^{n-1}} |\rho_K(\xi)-\rho_L(\xi)|.$$ If the Minkowski functional of a star body $K$ in $\R^{2n}$ is $R_\theta$-invariant (i.e. satisfies equations (\[rotation\])), we say that $K$ is a [*complex star body*]{} in $\R^{2n}.$

For $\xi\in \C^n,
|\xi|=1,$ denote by $$H_\xi = \{ z\in \C^n:\ (z,\xi)=\sum_{k=1}^n z_k\overline{\xi_k} =0\}$$ the complex hyperplane through the origin, perpendicular to $\xi.$ Under the standard mapping from $\C^n$ to $\R^{2n}$ the hyperplane $H_\xi$ turns into a $(2n-2)$-dimensional subspace of $\R^{2n}$ orthogonal to the vectors $$\xi=(\xi_{11},\xi_{12},...,\xi_{n1},\xi_{n2})\quad  {\rm and } \quad
 \xi^\perp =(-\xi_{12},\xi_{11},...,-\xi_{n2},\xi_{n1}).$$ The orthogonal two-dimensional subspace $H_\xi^\bot$ has orthonormal basis $\left\{\xi,\xi^\bot\right\}.$ A star (convex) body $K$ in $\R^{2n}$ is a complex star (convex) body if and only if, for every $\xi\in S^{2n-1},$ the section $K\cap H_\xi^\bot$ is a two-dimensional Euclidean circle with radius $\rho_K(\xi)= \|\xi\|_K^{-1}.$

