Balanced Complex Polytopes and Related Vector and Matrix Norms

We study the notion of balanced complex polytope as a generalization of a symmetric real polytope to the complex space $\mathbb{C}^n$. We pay particular attention to the geometric properties of such complex polytopes and of their counterparts in the adjoint form. In particular, we stress the differences occurring with respect to the well-known real case. We also introduce and discuss the related definitions of complex polytope norm and adjoint complex polytope norm