The strong Viterbo conjecture and various flavours of duality in Lagrangian products

Abstract: In this note we analyze normalized symplectic capacities for two different notions of duality in Lagrangian products. Let $\Phi$ be a $n$-tuple of Young functions with Legendre transform $n$-tuple $\Phi^*$ and $K_{\Phi}$ the unit ball for the Luxemburg metric induced by $\Phi$. We can consider the ``dual functional" Lagrangian product $K_{\Phi}\times_LK_{\Phi^*}$ and the usual polar dual Lagrangian product $K_{\Phi}\times_L K_{\Phi}^{\circ}$. We show that for the former, all normalized symplectic capacities agree, while for the latter, we give a lower bound depending on $\Phi$. In particular, under certain conditions on the $n$-tuple $\Phi$, we get that $c(K_{\Phi}\times_L K_{\Phi}^{\circ})=4$, for any normalized symplectic capacity, that is, the strong Viterbo conjecture holds.