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 $$\mathcal{E}:=\left\{(x,y)\in \R^2, x=\frac{4\pi A(\Omega)}{P(\Omega)^2},y=\frac{E(\Omega)P(\Omega)}{2\pi^2},\,\Omega\mbox{convex} \right\},$$ 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: $$\min_{\Omega\in\mathcal{C}}\{E(\Omega)+\mu A(\Omega)\},$$ 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.