# Polar set

Given a set S in ${\displaystyle \mathbb {R} ^{n}}$, its polar set is defined as the point set ${\displaystyle S^{\circ }=\bigcap _{x\in S}H(x)}$ where ${\displaystyle H(x)}$ is the set of all points ${\displaystyle y}$ such that ${\displaystyle x\cdot y\leq 1}$ using the dot product. There are more general definitions of the polar in study of vector spaces (in which case the polar set generally lies in a different vector space, often the dual space), but we will focus on ${\displaystyle \mathbb {R} ^{n}}$.
The polar set is always convex, closed, and contains the origin. If ${\displaystyle S}$ is closed, convex, bounded, and contains the origin, then ${\displaystyle S^{\circ \circ }=S}$. Also, it always holds that ${\displaystyle ({\text{conv}}\,S)^{\circ }=S^{\circ }}$ where "conv" denotes the closed convex hull.
For convex polytopes it is related to the dual, but is not identical, as it is sensitive to translation. It is equivariant with respect to rotations about the origin, and uniformly scaling S by a factor of k uniformly scales ${\displaystyle S^{\circ }}$ by 1/k.