---
tag: F39N9NBFPZ
kind: concept
name: Lattice polytope
msc: [52B20, 11P21]
specializes: [300K6CB1KH]
sources:
  - url: https://en.wikipedia.org/wiki/Integral_polytope
same_as:
  wikidata: Q5166519
  wikipedia: Integral polytope
---

## Summary

A lattice polytope, also called an integral polytope, is a [[convex polytope|300K6CB1KH]] whose vertices all have integer coordinates.
Counting the integer points in such a polytope and in its dilates leads to the [[Ehrhart polynomial|9DEG0WRDHR]].
Lattice polytopes link convex geometry with combinatorics, with integer programming and, through their [[normal fans|XJ83PTS3H0]], with [[toric varieties|YWB981MEF2]].

## Explanations

### L1 human

Draw a square grid of points and join some of the grid points to make a convex shape with straight sides.
If every corner of the shape sits on a grid point, the shape is a lattice polygon.
The square with corners $(0,0)$, $(2,0)$, $(2,2)$, $(0,2)$ is one; it contains $9$ grid points.
Lattice polytopes are the same idea in any dimension.