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

Barabanov norms have been introduced in Barabanov (1988) and constitute an important instrument to analyze 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 (2010)).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 (2005) and Guglielmi and Protasov (2013) - 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 introducedto compute the lower spectral radius of a finite family of matrices having an invariant cone.