Coarse distance from dynamically convex to convex

TL;DR

This paper introduces new examples of dynamically convex domains in four-dimensional space that are not symplectically convex and demonstrates they can be arbitrarily distant from convex domains using a novel numerical criterion.

Contribution

The authors provide new examples of non-symplectically convex, dynamically convex domains without relying on previous criteria and establish their large distance from convex domains in symplectic space.

Findings

New examples of non-symplectically convex, dynamically convex domains in R^4.

Domains can be arbitrarily far from convex domains in the symplectic Banach-Mazur distance.

Explicit numerical criterion for symplectic non-convexity.

Abstract

Chaidez and Edtmair have recently found the first example of dynamically convex domains in $\mathbb R^4$ that are not symplectomorphic to convex domains (called symplectically convex domains), answering a long-standing open question. In this paper, we discover new examples of such domains without referring to Chaidez-Edtmair's criterion. We also show that these domains are arbitrarily far from the set of symplectically convex domains in $\mathbb R^4$ with respect to the coarse symplectic Banach-Mazur distance by using an explicit numerical criterion for symplectic non-convexity.