# Minimum area of the symmetric difference of odd number of translated copies of a unit circle $C$

Let $$C$$ be the unit circle in a plane. Take an odd number $$n$$ of translated copies of $$C$$ and take their symmetric difference $$D$$. Is it true that the area of $$D$$ should be at least that of $$C$$?

If $$C$$ is a centrally symmetric convex 4-gon or 6-gon, then I can show that the statement is true. Take a lattice $$\Lambda$$ so that $$C$$ becomes the fundamental domain. Then for any point $$p$$ in the plane, $$p + \Lambda$$ intersects any translated copy of $$C$$ exactly once, so $$p + \Lambda$$ should hit at least one point of $$D$$ by parity. Now integrate $$p$$ over $$C$$ to conclude $$|D| \geq |C|$$.

I actually conjecture that the statement is true for any centrally symmetric and convex body $$C$$. Dropping any of such conditions, I have a corresponding counterexample (that is, centrally symmetric yet non-convex, or convex yet not centrally symmetric).

This seems to be an open problem:

Rom Pinchasi, On the odd area of the unit disc, Israel Journal of Mathematics 256, 619-637. https://doi.org/10.1007/s11856-023-2518-4

Amir Carmel and Rom Pinchasi, Some notes about the odd area of unit discs centered at points on a circle, Australas. J. Combin. 87 (2023), 148–159. https://ajc.maths.uq.edu.au/pdf/87/ajc_v87_p148.pdf