We investigate the convexity up to symplectomorphism (called symplectic convexity) of star-shaped toric domains in R 4. In particular, based on the criterion from Chaidez-Edtmair in [4] 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 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.