  We look for the minimizers of the functional $\jla{\la}(\oo)=\la|\oo|-P(\oo)$
among planar convex domains constrained to lie into a given ring. We prove
that, according to the values of the parameter $\la$, 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.
