Symplectic capacity and short periodic billiard trajectory

TL;DR

This paper proves that any smooth bounded domain in Euclidean space has a short periodic billiard trajectory with a limited number of bounces, using symplectic capacity methods to improve previous length bounds.

Contribution

It establishes a new upper bound on the length of periodic billiard trajectories based on the domain's inradius, utilizing symplectic homology techniques.

Findings

Existence of short periodic billiard trajectories with at most n+1 bounces.

Length of trajectories bounded by a constant times the inradius of the domain.

Improvement over previous volume-based length bounds.

Abstract

We prove that a bounded domain $\Omega$ in $\R^n$ with smooth boundary has a periodic billiard trajectory with at most $n+1$ bounce times and of length less than $C_n r(\Omega)$, where $C_n$ is a positive constant which depends only on $n$, and $r(\Omega)$ is the supremum of radius of balls in $\Omega$. This result improves the result by C.Viterbo, which asserts that $\Omega$ has a periodic billiard trajectory of length less than $C'_n \vol(\Omega)^{1/n}$. To prove this result, we study symplectic capacity of Liouville domains, which is defined via symplectic homology.