Symplectic Banach-Mazur distances between subsets of C^n

Abstract

Following proposals of Ostrover and Polterovich, we introduce and study "coarse" and "fine" versions of a symplectic Banach-Mazur distance on certain open subsets of $\mathbb{C}^n$ and other open Liouville domains. The coarse version declares two such domains to be close to each other if each domain admits a Liouville embedding into a slight dilate of the other; the fine version, which is similar to the distance on subsets of cotangent bundles of surfaces recently studied by Stojisavljević and Zhang, imposes an additional requirement on the images of these embeddings that is motivated by the definition of the classical Banach-Mazur distance on convex bodies. Our first main result is that the coarse and fine distances are quite different from each other, in that there are sequences that converge coarsely to an ellipsoid but diverge to infinity with respect to the fine distance. Our other main result is that, with respect to the fine distance, the space of star-shaped domains in $\mathbb{C}^n$ admits quasi-isometric embeddings of $\mathbb{R}^D$ for every finite dimension $D$. Our constructions are obtained from a general method of constructing $(2n+2)$-dimensional Liouville domains whose boundaries have Reeb dynamics determined by certain autonomous Hamiltonian flows on a given $2n$-dimensional Liouville domain. The bounds underlying our main results are proven using filtered equivariant symplectic homology via methods from prior joint work with Gutt.