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

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 capacities constructed authors based on a class Hamiltonian non-periodic boundary value problems recently. Then introduce billiards domains, and them prove some corresponding results those periodic obtained 2012.