Coarse distance from dynamically convex to convex

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.