Polar polytope

Summary

Let $P$ be a convex polytope with the origin in its interior. Its polar polytope is the set $P^\circ$ of vectors $y$ with $x \cdot y \le 1$ for all $x \in P$. It is again a convex polytope, and taking the polar twice gives back $P$. Polarity reverses the order of faces: vertices of $P$ correspond to facets of $P^\circ$ and facets of $P$ to vertices of $P^\circ$. It turns simple polytopes into simplicial polytopes and back, and it is the geometric form of the dual polyhedron.