On roundness of rotation sets

For a compact set L we define the roundness of L as the ratio between the area of L and the smallest round disk containing L. 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 ρ , where the parameter ρ ranges over irrational numbers in (0, 1). Each F ρ is a Kwapisz-like diffeomorphism with a 2-dimensional non-polygonal rotation set whose extreme point set contains exactly four (two-sided) accumulation points. We define the roundness of such a set as the ratio between its area and the area bounded by the circle of radius Ρ, and give its upper and lower bounds in terms of ρ. It is neither monotone nor continuous.