  We investigate the convexity up to symplectomorphism (called symplectic
convexity) of star-shaped toric domains in $\mathbb R^4$. In particular, based
on the criterion from Chaidez-Edtmair via Ruelle invariant and systolic ratio
of the boundary of star-shaped toric domains, we provide elementary operations
on domains that can kill the symplectic convexity. These operations only result
in $C^0$-small perturbations in terms of domains' volume. Moreover, one of the
operations is a systematic way to produce examples of dynamically convex but
not symplectically convex toric domains. Finally, we are able to provide
concrete bounds for the constants that appear in Chaidez-Edtmair's criterion.
