Shortest Closed Billiard Trajectories in the Plane and Equality Cases in Mahler's Conjecture

TL;DR

This paper investigates shortest closed billiard trajectories in the plane, establishing inequalities and properties related to Mahler's conjecture, with implications for symplectic geometry and convex analysis.

Contribution

It proves Rogers-Shepard type inequalities for billiard trajectories and explores their properties in Hanner polytopes relevant to Mahler's conjecture.

Findings

Established inequalities for shortest billiard trajectories in the plane

Identified properties of billiard trajectories in Hanner polytopes

Contributed to the symplectic approach to Mahler's conjecture

Abstract

In this note we prove some Rogers-Shepard type inequalities for the lengths of shortest closed billiard trajectories, mostly in the planar case. We also establish some properties of closed billiard trajectories in Hanner polytopes, having some significance in the symplectic approach to the Mahler conjecture.