Finding extremal complex polytope norms for families of real matrices

In this paper we consider finite families F of real nxn matrices. In particular, wefocus on the computation of the joint spectral radius r(F) 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 ofefficient 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 existenceof 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 orderto provide a more satisfactory treatment of the real case.