On roundness of rotation sets

Motivated by the question whether a round disk can be realized as the rotation set of a torus diffeomorphism, we study the roundness of rotation sets of a parametric family of torus diffeomorphisms $F_{\rho }$, where the parameter $\rho$ ranges over irrational numbers in $(0,1)$. Each $F_{\rho }$ is a Kwapisz-like diffeomorphism with a 2-dimensional non-polygonal rotation set $${\mathrm{\Lambda }}_{\rho }^{\mathrm{\prime }}=\mathrm{conv}\left(\{(\pm \frac{\lceil m\rho \rceil }{m+n+1},\pm \frac{\lceil n\rho \rceil }{m+n+1}):m,n\in {\mathbb{N}}_0,\lceil m\rho \rceil -m\rho <\rho ,\lceil n\rho \rceil -n\rho <\rho \}\right)$$ whose extreme point set contains exactly four (two-sided) accumulation points. We define the roundness of ${\mathrm{\Lambda }}_{\rho }^{\mathrm{\prime }}$ as the ratio $R_{\rho }=\frac{\mathrm{Area}({\mathrm{\Lambda }}_{\rho }^{\mathrm{\prime }})}{\pi {\rho }^2}$, and give its upper and lower bounds in terms of $\rho$. $R_{\rho }$ is neither monotone nor continuous.