Generalizations of Ekeland-Hofer and Hofer-Zehnder symplectic capacities and applications

This is the first installment in a series of papers aimed at generalizing symplectic capacities and homologies. The main purposes of this paper are to construct analogues of Ekeland-Hofer and Hofer-Zehnder symplectic capacities based on a class of Hamiltonian boundary value problems motivated by Clarke's and Ekeland's work, and to study generalizations of some important results about the original these two capacities (for example, the famous Weinstein conjecture, representation formula for $c_{\rm EH}$ and $c_{\rm HZ}$, a theorem by Evgeni Neduv, Brunn-Minkowski type inequality and Minkowski billiard trajectories proposed by Artstein-Avidan-Ostrover).Comment: Latex, 89 pages. Results in Section 1.6 are improved. Some typos are corrected. arXiv admin note: text overlap with arXiv:1903.0067