Finding extremal complex polytope norms for families of real matrices

In this paper we consider finite families F of real n×n matrices. In particular, we focus on the computation of the joint spectral radius via the detection of an extremal norm in the class of complex polytope norms, whose unit balls are balanced complex polytopes with a finite essential system of vertices. Such a finiteness property is very useful in view of the construction of efficient computational algorithms. More precisely, we improve the results obtained in our previous paper [N. Guglielmi, F. Wirth, and M. Zennaro, SIAM J. Matrix Anal. Appl., 27 (2005), pp. 721–743], where we gave some conditions on the family F which are sufficient to guarantee the existence of an extremal complex polytope norm. Unfortunately, they exclude unnecessarily many interesting cases of real families. Therefore, here we relax the conditions given in our previous paper in order to provide a more satisfactory treatment of the real case.