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.

中文翻译：

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凸域扩展辛容量的 Brunn-Minkowski 型不等式和一类台球轨迹的长度估计

在本文中，我们首先将 Artstein-Avidan-Ostrover 于 2008 年建立的 Ekeland-Hofer-Zehnder 有界凸域辛容量的 Brunn-Minkowski 型不等式推广到作者基于一类哈密​​顿非周期边值问题最近。然后我们引入了一类凸域非周期性台球，并为它们证明了一些与 Artstein-Avidan-Ostrover 在 2012 年获得的凸域中周期性台球的结果相对应的结果。