Normal fan

Summary

The normal fan of a convex polytope $P$ is the collection of its normal cones, one cone for each face. The normal cone of a face collects the linear functionals that reach their maximum on $P$ at that face. The normal fan records the combinatorics of $P$ in a dual form. When $P$ is full-dimensional, its vertices give the full-dimensional cones and its facets give the rays. For a full-dimensional lattice polytope it is a rational fan, and it defines the projective toric variety of the polytope.