**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.