Polar set

Given a set S in $$\mathbb{R}^n$$, its polar set is defined as the point set $$S^\circ = \bigcap_{x \in S} H(x, 1)$$ where $$H(x)$$ is the set of all points $$y$$ such that $$x \cdot y \leq 1$$ using the dot product. There are more general definitions of S in study of vector spaces, but we will focus on $$\mathbb{R}^n$$.

The polar set is always convex and closed. For convex polytopes it is related to the dual, but is not identical, as it is sensitive to translation (but it is equivariant with respect to uniform scaling and rotation about the origin).