Irie, K. Math. Z. (2012) 272: 1291. doi:10.1007/s00209-012-0987-y

Abstract

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