On a question of Gromov about the Wirtinger inequalities (Q1663779)

This paper studies various inequalities on metric spaces: Gromov's \(\mathrm{Cycl}_k(0)\) conditions and the Wirtinger inequalities \(\mathrm{Wir}_k\). Gromov showed that for any \(k \geq 4\), the property of being \(\mathrm{CAT}(0)\) implies \(\mathrm{Cycl}_k(0)\), which in turn implies \(\mathrm{Wir}_k\). On the other hand, for a geodesic space, the \(\mathrm{Cycl}_4(0)\) condition implies \(\mathrm{CAT}(0)\), and so for geodesic spaces the implication \(\mathrm{Cycl}_4(0) \implies \mathrm{Wir}_k\) holds for all \(k \geq 4\). Does this statement, \(\mathrm{Cycl}_4(0) \implies \mathrm{Wir}_k\), hold without the assumption that the space is geodesic? This paper answers this question of Gromov in the affirmative, using Perron-Frobenius theory.