A Dido problem for domains in \(\mathbb R^ 2\) with a given inradius (Q914155)

Let \(H\) denote the half-plane \(y\geq 0\) in \(\mathbb R^ 2\) and let \(D\subset H\) be a simply connected compact domain whose boundary \(\partial D\) is a rectifiable simple curve and put \(\gamma =\partial D\cap \partial H\), \(\Gamma =\partial D-\gamma\). The authors solve the problem of finding the domain \(D\) of given inradius \(\rho\) and area \(A\) which minimizes the length of \(\Gamma\). Assuming \(\rho =1\), and therefore \(A>\pi\), the authors distinguish three cases according to the value of \(A\): \(\pi <A<(\pi /2)+2\), \((\pi /2)+2<A<2\pi,\) \(2\pi <A\). The solution is not convex in the first case and convex in the other two cases. The proof uses Steiner symmetrization and some suitable known geometric inequalities.