I was looking for a characterization of compact subsets of $\mathbb{R}^2$ that are stable under a central symmetry.
Let $s_M$ denote the symmetry of center $M$.
If $A$ is such a set that is measurable with $\mu(A)=1$, then there is a center $M$ such that $A=s_M(A)$, hence $\mu(A \cap s_M(A))=1$.
Now, if a set is not stable under a central symmetry, is it true that for any point $M$, one has $\mu(A \cap s_M(A))<1$? By a compactness argument, this would imply the existence of a constant term $0<C<1$ such that $\underset{M \in \mathbb{R}^2}{\max}(\mu(A \cap s_M(A)))=C$.
(Say differently : take a compact set of measure 1. Take a picture of it, do a half-turn rotation, and translate it such that the matching part has maximal measure).
I wonder how to find sets with the lowest possible constant. Is it studied somewhere?
Thanks in advance for your comments :)