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The problem of construction of Barabanov norms for analysis of properties ofthe joint (generalized) spectral radius of matrix sets has been discussed in anumber of publications. In previous papers of the author the method ofBarabanov norms was the key instrument in disproving the Lagarias-WangFiniteness Conjecture. The related constructions were essentially based on thestudy of the geometrical properties of the unit balls of some specificBarabanov norms. In this context the situation when one fails to find amongcurrent publications any detailed analysis of the geometrical properties of theunit balls of Barabanov norms looks a bit paradoxical. Partially this isexplained by the fact that Barabanov norms are defined nonconstructively, by animplicit procedure. So, even in simplest cases it is very difficult tovisualize the shape of their unit balls. The present work may be treated as thefirst step to make up this deficiency. In the paper an iteration procedure isconsidered that allows to build numerically Barabanov norms for the irreduciblematrix sets and simultaneously to compute the joint spectral radius of thesesets.

Max-relaxation iteration procedure for building of Barabanov norms: convergence and examples