Elastic energy of a convex body

In this paper a Blaschke‐Santaló 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 : = ( x , y ) ∈ R 2 , x = 4 π A ( Ω ) P ( Ω ) 2, y = E ( Ω ) P ( Ω ) 2 π 2, Ω convex , 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 Ω ∈ C{ E ( Ω ) + μ A ( Ω ) } , 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.