Complex polytope extremality results for families of matrices

In this paper we consider finite families of complex n × n-matrices. In particular, we focus on those families that satisfy the so-called finiteness conjecture, which was recently disproved in its more general formulation. We conjecture that the validity of the finiteness conjecture for a finite family of nondefective type is equivalent to the existence of an extremal norm in the class of complex polytope norms. However, we have not been able to prove this complex polytope extremality conjecture, but we are able to prove the small complex polytope extremality theorem under some more restrictive hypotheses on the underlying family of matrices. In addition, our theorem assures a certain finiteness property on the number of vertices of the unit ball of the extremal complex polytope norm, which could be very useful for the construction of suitable algorithms aimed at the actual computation of the spectral radius of the family.