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).
