Viterbo's conjecture for Lagrangian products in $\mathbb{R}^4$ and symplectomorphisms to the Euclidean ball

TL;DR

This paper proves Viterbo's conjecture for certain 4D Lagrangian products, classifies equality cases, and explores symplectomorphisms to Euclidean balls, with implications for billiard inequalities and symplectic flow properties.

Contribution

It extends Viterbo's conjecture verification to new Lagrangian product classes and classifies equality cases, linking them to symplectomorphisms and billiard inequalities.

Findings

Viterbo's conjecture holds for specific 4D Lagrangian products.

Most equality cases are symplectomorphic to Euclidean balls.

Flow of equality cases is not Zoll, but satisfies a weaker Zoll property.

Abstract

We use the generalized Minkowski billiard characterization of the EHZ-capacity of Lagrangian products in order to reprove that the $4$-dimensional Viterbo conjecture holds for the Lagrangian products (any triangle/parallelogram in $\mathbb{R}^2$)$\times$(any convex body in $\mathbb{R}^2$) and extend this fact to the Lagrangian products (any trapezoid in $\mathbb{R}^2$)$\times$(any convex body in $\mathbb{R}^2$). Based on this analysis, we classify equality cases of this version of Viterbo's conjecture and prove that most of them can be proven to be symplectomorphic to Euclidean balls. As a by-product, we prove sharp systolic Minkowski billiard / worm problem inequalities. Furthermore, we discuss the Lagrangian products (any convex quadrilateral in $\mathbb{R}^2$)$\times$(any convex body in $\mathbb{R}^2$) for which we show that the truth of Viterbo's conjecture would follow from the positive solution of a challenging Euclidean covering problem. Finally, we show that the flow associated to equality cases of Viterbo's conjecture for Lagrangian products in $\mathbb{R}^4$--which turn out to be convex polytopes--is not Zoll in general, but that a weaker Zoll property, namely, that every characteristic almost everywhere away from lower-dimensional faces is closed and action-minimizing, does apply.