Chaidez and Edtmair have recently found the first examples of dynamically convex domains in R4 that are not symplectomorphic to convex domains (called symplectically convex domains), answering a long-standing open question. In this talk we shall present new examples of such domains without referring to Chaidez-Edtmair’s criterion. We shall show that these domains are arbitrarily far from the set of symplectically convex domains in R4  with respect to the coarse symplectic Banach-Mazur distance by using an explicit numerical criterion for symplectic non-convexity. This is joint work with J.Dardennes, V.Ramos and J.Zhang