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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."},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1010.3170","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SG","submitted_at":"2010-10-15T13:58:28Z","cross_cats_sorted":["math.DS"],"title_canon_sha256":"c0832a32b687a8117abc78540cea3e117867172d57ad5275f44410be31aba5c4","abstract_canon_sha256":"7f0bca74717b1fbda74141e2ee91f7c5c6f46ec54bab7853cf3f6a257a201fc9"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:03:07.267054Z","signature_b64":"ZxQDHP51S2YePm+wuJl6jFqxxqKZSUY9awqasYearGTm0D8vuz9DYdG5qwv9XJNwv5wE0IoAXiqPJWhKNOpUBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"501f36c9f2a47b95709fcfd147d1dcf34c55aa5c26b866d03f72dc66b72dc531","last_reissued_at":"2026-05-18T04:03:07.266388Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:03:07.266388Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Symplectic capacity and short periodic billiard trajectory","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.DS"],"primary_cat":"math.SG","authors_text":"Kei Irie","submitted_at":"2010-10-15T13:58:28Z","abstract_excerpt":"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 $\\Omega$. 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