A Brunn-Minkowski type inequality for extended symplectic capacities of convex domains and length estimate for a class of billiard trajectories

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.

Latex, 29 pages. Final version. To appear in Abhandlungen aus dem Mathematischen Seminar der Universitat Hamburg Abh.Math. Semin. Univ.Hambg. The preprint arXiv:1903.01116[math.SG] was split into two papers. This is one of them. arXiv admin note: substantial text overlap with arXiv:1903.01116