  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.
