Elastic Energy of a convex body

In this paper a Blaschke-Santal'o diagram involving the area, the perimeter and the elastic energy of planar convex bodies is considered. In order to do this, we investigate the following shape optimization problem: minΩ∈C{E(Ω)+μA(Ω)}, where C is the class of convex bodies with fixed perimeter and μ≥0 is a parameter, A is the area and E is the elastic energy, that is a Willmore type energy in the plane. Existence, regularity and geometric properties of solutions to this minimum problem are shown.