no systematic studies of these bodies have been carried out, and results appear only occasionally; see for example [@KKZ; @KZ; @AB; @Ru1; @Zy; @Zy1].

origin symmetric convex bodies in $\C^n$ are the unit balls of norms on $\C^n.$ We denote by $\|\cdot\|_K$ the norm corresponding to the body $K:$ $$K=\{z\in \C^n:\ \|z\|_K\le 1\}.$$ In order to define volume, we identify $\C^n$ with $\R^{2n}$ using the standard mapping $$\xi = (\xi_1,...,\xi_n)=(\xi_{11}+i\xi_{12},...,\xi_{n1}+i\xi_{n2})
 \mapsto  (\xi_{11},\xi_{12},...,\xi_{n1},\xi_{n2}).$$ Since norms on $\C^n$ satisfy the equality $$\|\lambda z\| = |\lambda|\|z\|,\quad \forall z\in \C^n,\  \forall\lambda \in \C,$$ origin symmetric complex convex bodies correspond to those origin symmetric convex bodies $K$ in $\R^{2n}$ that are invariant with respect to any coordinate-wise two-dimensional rotation, namely for each $\theta\in [0,2\pi]$ and each $\xi= (\xi_{11},\xi_{12},...,\xi_{n1},\xi_{n2})\in \R^{2n}$ $$\label{rotation}
  \|\xi\|_K =
 \|R_\theta(\xi_{11},\xi_{12}),...,R_\theta(\xi_{n1},\xi_{n2})\|_K,$$ where $R_\theta$ stands for the counterclockwise rotation of $\R^2$ by the angle $\theta$ with respect to the origin. We shall say that $K$ is a [*complex convex body in $\R^{2n}$*]{} if $K$ is a convex body and satisfies equations (\[rotation\]).

A compact set $K$ in $\R^n$ is called a star body if the origin is an interior point of $K$, every straight line passing through the origin crosses the boundary of $K$ 