---
tag: 300K6CB1KH
kind: concept
name: Convex polytope
msc: [52B11]
specializes: [02AS45T3H9]
sources:
  - url: https://en.wikipedia.org/wiki/Convex_polytope
same_as:
  wikidata: Q4129097
  wikipedia: Convex polytope
---

## Summary

A convex polytope is the [[convex hull|0FQSDC2EQ2]] of finitely many points of $\R^n$.
Equivalently, it is a bounded set cut out by finitely many linear inequalities.
Convex polygons, convex polyhedra such as the cube and the regular tetrahedron, and their analogues in higher dimensions are convex polytopes.
They are the basic objects of polyhedral combinatorics and of linear programming, where the feasible region of a bounded problem is a convex polytope.

## Explanations

### L1 human

Hammer a few nails into a board and stretch a rubber band around them all: the region inside the band is a convex polygon.
A convex polytope is the same idea in any dimension: the smallest convex set containing a finite list of points.
It has flat sides, straight edges and finitely many corners.
For example, the unit square is the convex hull of the four points $(0, 0)$, $(1, 0)$, $(0, 1)$ and $(1, 1)$.