Optimal sets for a class of minimization problems with convex constraints

We look for the minimizers of the functional $\text{jla}\text{la}(\text{oo})=\text{la}\mathrm{\mid }\text{oo}\mathrm{\mid }-P(\text{oo})$ among planar convex domains constrained to lie into a given ring. We prove that, according to the values of the parameter $\text{la}$, the solutions are either a disc or a polygon. In this last case, we describe completely the polygonal solutions by reducing the problem to a finite dimensional optimization problem. We recover classical inequalities for convex sets involving area, perimeter and inradius or circumradius and find a new one.