Abstract: We discuss two isoperimetric problems in the plane. The first one concerns N identical point interactions placed at vertices of an equilateral polygon, in the second case we have a delta interaction supported by a smooth loop. We ask about the geometric configuration which maximizes the ground state energy and find it is locally achieved in case with maximum symmetry, i.e. a regular polygon and a circle, respectively. On the way we find that the problem can be reformulated in terms of inequalities about mean chord values which lead us to several open problems.