---
tag: 7E6FXTGH2Y
kind: concept
name: Simple polytope
msc: [52B05, 52B11, 52B12]
specializes: [300K6CB1KH]
sources:
  - url: https://en.wikipedia.org/wiki/Simple_polytope
same_as:
  wikidata: Q4381515
  wikipedia: Simple polytope
---

## Summary

A simple polytope is a [[convex polytope|300K6CB1KH]] of dimension $d$ in which every [[vertex|R7V3S9V891]] lies on exactly $d$ edges.
Near each vertex it then looks like the corner of a cube.
Simple polytopes are the duals of [[simplicial polytopes|YVJ15PGKDM]], so results about one class translate into results about the other.
Polytopes given by inequalities in general position are simple, which makes them the generic case in linear programming.

## Explanations

### L1 human

At each corner of a cube exactly three edges meet, and the cube lives in three dimensions.
A polytope with this property at every corner, in any dimension, is called simple.
The square pyramid is not simple: four edges meet at its apex.
If you slightly shift the walls of any polytope in a random way, the result is simple.