Symplectic capacity and short periodic billiard trajectory

Abstract

We prove that a bounded domain Ω 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(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′nvol(Ω)^[1/n] . To prove this result, we study symplectic capacity of Liouville domains, which is defined via symplectic homology.