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

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