Asymptotic equivalence of symplectic capacities

Abstract

A long-standing conjecture states that all normalized symplectic capacities coincide on the class of convex subsets of ${\mathbb{R}}^{2n}$. In this note we focus on an asymptotic (in the dimension) version of this conjecture, and show that when restricted to the class of centrally symmetric convex bodies in ${\mathbb{R}}^{2n}$, several symplectic capacities, including the Ekeland–Hofer–Zehnder capacity, the displacement energy capacity, and the cylindrical capacity, are all equivalent up to a universal constant.