Geometrical aspects of expansions in complex bases

We study the set of the representable numbers in base $q=p{e}^{i\frac{2\pi }{n}}$ with $\rho >1$ and $n\in \mathbb{N}$ and with digits in a arbitrary finite real alphabet $A$. We give a geometrical description of the convex hull of the representable numbers in base $q$ and alphabet $A$ and an explicit characterization of its extremal points. A characterizing condition for the convexity of the set of representable numbers is also shown.