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 E:=left(x,y)∈ R2, x=frac4π A(Omega)P(Omega)2,y=fracE(Omega)P(Omega)2π2,,Omegamboxconvex 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: minOmega∈CE(Omega)+μ A(Omega), where C is the class of convex bodies with fixed perimeter and μ≥ 0 is a parameter. Existence, regularity and geometric properties of solutions to this minimum problem are shown.