Let K be a compact centro-symmetric subset of the Euclidean plane. Assuming that K has constant "mass density" at all of its points, does the center of gravity-also called the "centroid"-of K, always coincide with the symmetric center of K? Intuitively, I would say "yes". But what about cases where no straight line L in the plane is an "axis of symmetry" of K? By this, I mean that K would be mapped into itself by orthogonal reflection across L.
Let $O$ be the center of symmetry of $K$, and let $G$ be its centroid.
Consider now a point reflection across $O$ which transforms $K$ to $K'$ and $G$ to $G'$. Since the point reflection is an affine transformation and affine transformations preserve centroids, $G'$ must be the centroid of $K'$.
Given that $K$ is centrally symmetric with respect to $O$, it follows that $K' \equiv K$ and therefore $G' \equiv G$. But the only fixed point of a point reflection is its center, so in the end $G \equiv O$.