  In this paper we study a constrained minimization problem for the Willmore
functional. For prescribed surface area we consider smooth embeddings of the
sphere into the unit ball. We evaluate the dependence of the the minimal
Willmore energy of such surfaces on the prescribed surface area and prove
corresponding upper and lower bounds. Interesting features arise when the
prescribed surface area just exceeds the surface area of the unit sphere. We
show that (almost) minimizing surfaces cannot be a $C^2$-small perturbation of
the sphere. Indeed they have to be nonconvex and there is a sharp increase in
Willmore energy with a square root rate with respect to the increase in surface
area.
