Abstract
In this paper, we firstly generalize the Brunn–Minkowski type inequality for Ekeland–Hofer–Zehnder symplectic capacity of bounded convex domains established by Artstein-Avidan–Ostrover in 2008 to extended symplectic capacities of bounded convex domains constructed by authors based on a class of Hamiltonian non-periodic boundary value problems recently. Then we introduce a class of non-periodic billiards in convex domains, and for them we prove some corresponding results to those for periodic billiards in convex domains obtained by Artstein-Avidan–Ostrover in 2012.
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The preprint was split into two papers, which were submitted independently. The present paper is one of them, mainly consisting of contents in Sections 8, 9 of [13].
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We are deeply grateful to the anonymous referees for giving very helpful comments and suggestions to improve the exposition.
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Communicated by Janko Latschev.
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Jin, R., Lu, G. A Brunn–Minkowski type inequality for extended symplectic capacities of convex domains and length estimate for a class of billiard trajectories. Abh. Math. Semin. Univ. Hambg. 93, 1–30 (2023). https://doi.org/10.1007/s12188-023-00263-z
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DOI: https://doi.org/10.1007/s12188-023-00263-z