Center of symmetry ($S=2\mathbf{c}-S$)

A point $\mathbf{c}$ is a center of symmetry of a set $S$ in Euclidean space if point reflection $\mathbf{x}\mapsto 2\mathbf{c}-\mathbf{x}$ maps $S$ onto itself. Every ellipse has its center of an ellipse as a center of symmetry. More generally, a nonempty bounded set has at most one: two such reflections compose to translation by twice the difference of their centers, and iterating a nonzero translation contradicts boundedness.