The Minkowski Billiard Characterization of the EHZ-capacity of Convex Lagrangian Products

TL;DR

This paper establishes a general connection between the EHZ-capacity of convex Lagrangian products and Minkowski billiard trajectories, enabling computational approaches for polytopes without smoothness assumptions.

Contribution

It proves the Minkowski billiard characterization of EHZ-capacity for convex bodies without smoothness or convexity restrictions, extending previous results.

Findings

Connection between EHZ-capacity and Minkowski billiard trajectories established.

Enables computation of EHZ-capacity for convex polytopes using discrete methods.

Generalization of previous smoothness-dependent results.

Abstract

We rigorously state the connection between the EHZ-capacity of convex Lagrangian products $K\times T\subset\mathbb{R}^n\times\mathbb{R}^n$ and the minimal length of closed $(K,T)$-Minkowski billiard trajectories. This connection was made explicit for the first time by Artstein-Avidan and Ostrover under the assumption of smoothness and strict convexity of both $K$ and $T$. We prove this connection in its full generality, i.e., without requiring any conditions on the convex bodies $K$ and $T$. This prepares the computation of the EHZ-capacity of convex Lagrangian products of two convex polytopes by using discrete computational methods.