Combinatorial formulas for some generalized Ekeland-Hofer-Zehnder capacities of convex polytopes

TL;DR

This paper extends combinatorial formulas for Ekeland-Hofer-Zehnder capacities to generalized capacities of convex polytopes, revealing new subadditivity properties for coisotropic capacities under hyperplane cuts.

Contribution

It provides new combinatorial formulas for generalized Ekeland-Hofer-Zehnder capacities of convex polytopes, including coisotropic capacities, and explores their subadditivity properties.

Findings

Formulas for $ ext{ extPsi}$-Ekeland-Hofer-Zehnder capacities of convex polytopes.

Formulas for coisotropic Ekeland-Hofer-Zehnder capacities of convex polytopes.

Subadditivity of coisotropic capacities under hyperplane cuts in 2D.

Abstract

Motivated by Pazit Haim-Kislev's combinatorial formula for the Ekeland-Hofer-Zehnder capacities of convex polytopes, we give corresponding formulas for $\Psi$-Ekeland-Hofer-Zehnder and coisotropic Ekeland-Hofer-Zehnder capacities of convex polytopes introduced by the second named author and others recently. Contrary to Pazit Haim-Kislev's subadditivity result for the Ekeland-Hofer-Zehnder capacities of convex domains, we show that the coisotropic Hofer-Zehnder capacities satisfy the subadditivity for suitable hyperplane cuts of two-dimensional convex domains in the reverse direction.