On the odd area of the unit disc

Let ℱ be a family of n unit discs in the plane, where n is odd. It is by now a long standing open problem what is the minimum (over all such families ℱ) possible area OA(ℱ) of all the points in the plane that belong to an odd number of discs in ℱ . In this paper we develop necessary geometric conditions for a family ℱ such that OA(ℱ) is minimum. We apply this result to answer an interesting special case of the above mentioned problem.