Canonical construction of polytope Barabanov norms and antinorms for sets of matrices

Barabanov norms have been introduced in Barabanov (Autom. Remote Control, 49 (1988), pp. 152–157) and constitute an important instrument in analyzing the joint spectral radius of a family of matrices and related issues. However, although they have been studied extensively, even in very simple cases it is very difficult to construct them explicitly (see, e.g., Kozyakin (Discrete Contin. Dyn. Syst. Ser. B, 14 (2010), pp. 143–158)). In this paper we give a canonical procedure to construct them exactly, which associates a polytope extremal norm—constructed by using the methodologies described in Guglielmi, Wirth, and Zennaro (SIAM J. Matrix Anal. Appl., 27 (2005), pp. 721–743) and Guglielmi and Protasov (Found. Comput. Math., 13 (2013), pp. 37–97)—to a polytope Barabanov norm. Hence, the existence of a polytope Barabanov norm has the same genericity of an extremal polytope norm. Moreover, we extend the result to polytope antinorms, which have been recently introduced to compute the lower spectral radius of a finite family of matrices having an invariant cone.