Some notes about the odd area of unit discs centered at points on a circle

Let F be a family of n unit discs in the plane, where n is odd. A well-known open problem seeks to determine 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 show that if F is a family of n unit discs in the plane whose centers lie on a circle of radius r where 0 ≤ r ≤ 1, centered at the origin, then OA(F) ≥ π. Furthermore, we show that as we push the discs in F towards the origin, keeping their centers on a circle centered at the origin, the function OA(F) decreases. Additionally, we provide a separate proof for the interesting case of r = 1 using completely different ideas. One of the key tools we use is a new trigonometric inequality that is of independent interest. For a fixed odd integer n ≥ 3 and 0 ≤ α₀ ≤ α₁ ≤ … ≤ α_n−1 ≤ 2π, we show ∑ (−1)^j−i+1 sin(α_j − α_i) ≥ 0. 0≤i<j<n.