Minimizing clusters: existence and planar examples

The problem of enclosing a fixed area inside a figure in the plane with least perimeter was known since the times of ancient Greeks. They knew that the optimal solution was a circle, although they did not prove this fact precisely but just by approximation. Surprisingly, the first rigorous proof was found only in the 19th century. First Steiner showed that, if a solution exists, then it is necessarily a ball and, some years later, Carathéodory completed the proof showing the existence of the minimizers. We could generalize this problem, for example, trying to find two sets of fixed areas which minimize the perimeter of their boundary. In general, this problem could be set with N subsets of R n . This is called partitioning problem.