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

TL;DR

This paper investigates symplectic capacities in Lagrangian products, demonstrating conditions under which the strong Viterbo conjecture holds, especially for duality notions involving Young functions and their Legendre transforms.

Contribution

It establishes that all normalized symplectic capacities agree for the dual functional Lagrangian product and provides conditions where the strong Viterbo conjecture is verified.

Findings

All capacities agree for the dual functional Lagrangian product.

A lower bound on capacities depending on Young functions.

Conditions under which the strong Viterbo conjecture holds.

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.