New estimations for coisotropic Ekeland-Hofer-Zehnder capacity (Q6536755)

A symplectic capacity on the \(2n\)-dimensional Euclidean space \(\mathbb{R}^{2n}\) with the standard symplectic structure \(\omega_0\) is a map \(c\) which associates to each subset \(U\) of \(\mathbb{R}^{2n}\) a number \(c(U)\) satisfying some properties: monotonicity; conformality and nontriviality. Notice that the relevance of symplectic capacities is that they are symplectic invariants. \textit{I. Ekeland} and \textit{H. Hofer} [Math. Z. 200, No. 3, 355--378 (1989; Zbl 0641.53035)] and \textit{H. Hofer} and \textit{E. Zehnder} [Analysis, et cetera, Res. Pap. in Honor of J. Moser's 60th Birthd., 405--427 (1990; Zbl 0702.58021)] showed, that the Ekeland-Hofer capacity cEH(K) and the Hofer-Zehnder capacity cHZ(K) were both equal to \N\[\N\mathrm{cEHZ}(K) := \text{min } \{A(x) > 0 \mid x \text{ is a closed characteristic on } S \} ,\N\]\Nwhere \(K\) is a compact convex domain with smooth boundary \(S\). The above quantity is called the Ekeland-Hofer-Zehnder capacity. More recently, \textit{S. Lisi} and \textit{A. Rieser} [J. Symplectic Geom. 18, No. 3, 819--865 (2020; Zbl 1478.53126)] introduced the notion of a coisotropic capacity and constructed a coisotropic Hofer-Zehnder capacity, and \textit{R. Jin} and \textit{G. Lu} [J. Fixed Point Theory Appl. 25, No. 2, Paper No. 54, 59 p. (2023; Zbl 1521.53062)] constructed a relative version of the Ekeland-Hofer capacity with respect to a special class of coisotropic subspaces. The aim of the paper under review is just to obtain an estimation for coisotropic Ekeland-Hofer-Zehnder capacities by means of a combinatorial formula. The author also discusses the relationship between coisotropic Ekeland-Hofer-Zehnder capacities and Ekeland-Hofer-Zehnder capacities, a relevant question in symplectic geometry.