We find which are the simply connected domains in R 2 satisfying the Dido condition for a straight shoreline, with a given area A and a fixed inradius p, which minimize the length of the free boundary. There are three different cases according to the values of A and p.
1. GENERAL STATEMENT OF THE PROBLEM
Let 9 be a simply connected compact domain in R 2 whose boundary ~9 is a rectifiable simple curve and 9 ~ ~. We put 7 = 69 n 6~, F = ~9 -7. The constraint that we shall impose upon 9 is that it has fixed inradius, which without loss of generality, we can consider of unit length. We try to find which of such domains with a given area A minimize the length of F.
Let us assume that this problem has a solution 9o and let ~ be an incircle of 90. Then, by variational arguments it could be shown that the 'free' boundary of ~o, that is the part of ~9 o that does not touch 6~(e ~ or any incircle, must be the union of arcs of circles, all of them of the same curvature; and also that the free boundary meets ~ at right angles and 6~ tangentially. One can also get an intuitive feeling that this is true by thinking of 9 as a sort of two-dimensional liquid bounded by a tight elastic line F whose ends are in 6~(F and can slide freely along it; this freedom of movement is due to the fact that the 'shoreline' V is not considered in the boundary length. The constraint can be thought to be a rigid circle of unit radius inside 9 which F must enclose; the requirement that F should not enclose a greater circle is more difficult to represent physically.
In order to prove that the guessed figure is indeed the solution, we have to distinguish three cases according to the value of A: rc < A < (re/2)+ 2, (7~/2) + 2 < A < 2re, 2re < A. We want to remark that in the first two cases we could start from any compact domain 9, taking as length of F the outer Minkowski content of ~9 minus the Lebesgue measure of V. In fact, we shall make use only of Steiner symmetrization that does not increase the *Research partially supported by grant PS87-0115-C03-01 of the DGICYT.