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We address the problem of the computation of the spectral radius
 of a family of matrices. We briefly describe the extension of the
 concept of polytope to the complex space and outline the main
 geometric properties of such an object. Then we consider the norms
 determined by the complex polytopes and illustrate a possible
 algorithm for the approximation of the joint spectral radius of a
 family of matrices which is based on these complex polytope norms.
 As an example for our technique we consider the set of two
 matrices recently analyzed by Blondel, Nesterov and Theys to
 disprove the finiteness conjecture.

 This is a joint work with Marino Zennaro (Universit\`{a} di
 Trieste, Italy).
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