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

Abstract

AbstractWe rigorously state the connection between the EHZ-capacity of convex Lagrangian products $$K\times T\subset \mathbb {R}^n\times \mathbb {R}^n$$ K × T ⊂ R n × 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.