Counterexamples to the Complex Polytope Extremality Conjecture

We disprove a recent conjecture of Guglielmi, Wirth, and Zennaro, stating that any nondefective set of matrices having the finiteness property has an extremal complex polytope norm. We give two counterexamples that show that the conjecture is false even if the set of matrices is supposed to admit the positive orthant as an invariant cone, or even if the set of matrices is assumed to be irreducible.

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