Bounds for Minkowski Billiard Trajectories in Convex Bodies

TL;DR

This paper establishes bounds on the shortest periodic billiard trajectories within convex bodies using symplectic capacity, applicable to both classical and Minkowski billiards, advancing understanding of geometric and dynamical properties.

Contribution

It introduces new bounds for billiard trajectory lengths in convex bodies using symplectic capacity, extending results to Minkowski billiards.

Findings

Bounds for shortest billiard trajectories derived

Results apply to both classical and Minkowski billiards

Provides inequalities relating geometry and dynamics

Abstract

In this paper we use the Ekeland-Hofer-Zehnder symplectic capacity to provide several bounds and inequalities for the length of the shortest periodic billiard trajectory in a smooth convex body in ${\mathbb R}^{n}$. Our results hold both for classical billiards, as well as for the more general case of Minkowski billiards.