---
tag: XJ83PTS3H0
kind: concept
name: Normal fan
msc: [52B11, 14M25]
sources:
  - url: https://en.wikipedia.org/wiki/Toric_variety
---

## Summary

The normal fan of a [[convex polytope|300K6CB1KH]] $P$ 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|YWB981MEF2]] of the polytope.

## Explanations

### L1 human

Stand at a corner of a polytope and ask in which directions that corner is the farthest point of the polytope.
These directions form a cone, the normal cone of the corner.
Doing the same for edges and the other faces cuts the space of directions into cones that fit together with no gaps or overlaps.
For a square centred at the origin, the four corners give the four quadrants and the four sides give the four half-axes.