Elastic energy of a convex body

Following L. Euler, we define the elastic energy E(K) of a regular compact set K in the plane
as 1/2 times the integral over the boundary of K of the square of the boundary curvature. We will denote by $A(K)$ the area of $K$ and
$P(K)$ its perimeter. In this talk, we prove that for any convex set K the quotient
A(K)E(K)/P(K) is larger than or equal to pi/2, with equality only for the disk. We deduce that the disk
minimizes the elastic energy with an area constraint.
We will also consider analogous tridimensional problems involving the
Willmore (or the Helfrich) energy linked to the modelling of vesicles.