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. More precisely we give a description of set $$\mathcal{E}:=\{(x,y)\in {\mathbb{R}}^2,x=\frac{4\pi A(\mathrm{\Omega })}{P(\mathrm{\Omega }{)}^2},y=\frac{E(\mathrm{\Omega })P(\mathrm{\Omega })}{2{\pi }^2},\text{Omegaconvex}\},$$ where $A$ is the area, $P$ is the perimeter and $E$ is the elastic energy, that is a Willmore type energy in the plane. In order to do this, we investigate the following shape optimization problem: $$\underset{\mathrm{\Omega }\in \mathcal{C}}{\min }\{E(\mathrm{\Omega })+\mu A(\mathrm{\Omega })\},$$ where $\mathcal{C}$ is the class of convex bodies with fixed perimeter and $\mu \ge 0$ is a parameter. Existence, regularity and geometric properties of solutions to this minimum problem are shown.