Viterbo‘s conjecture for Lagrangian products in ℝ4
We show that Viterbo‘s conjecture (for the EHZ-capacity) for convex Lagrangian products in ℝ4 holds for all Lagrangian products (any trapezoid in ℝ2) x (any convex body in ℝ2). Moreover, we classify all equality cases of Viterbo’s conjecture within this configuration and show which of them are symplectomorphic to a Euclidean ball. As by-product, we conclude sharp systolic Minkowski billiard inequalities for geometries which have trapezoids as unit balls. Finally, we show that the flows associated to the above mentioned equality cases (which are polytopes) satisfy a weak Zoll property, namely, that every characteristic that is almost everywhere away from lower-dimensional faces is closed, runs over exactly 8 facets, and minimizes the action.