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Symplectic capacity and short periodic billiard trajectory

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 $C'_n \vol(Ω)^{1/n}$. To prove this result, we study symplectic capacity of Liouville domains, which is defined via symplectic homology.