Asymptotic Equivalence of Symplectic Capacities

TL;DR

This paper demonstrates that various symplectic capacities are asymptotically equivalent for centrally symmetric convex bodies in high-dimensional Euclidean spaces, supporting a long-standing conjecture in symplectic geometry.

Contribution

It establishes the asymptotic equivalence of multiple symplectic capacities for a specific class of convex bodies, advancing understanding of symplectic invariants.

Findings

Symplectic capacities are equivalent up to a constant for centrally symmetric convex bodies.

Supports the conjecture that all normalized symplectic capacities coincide on convex sets.

Provides a dimension-dependent perspective on symplectic capacity equivalences.

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 an absolute constant.