---
tag: W8P2G5R02N
kind: concept
name: H-representation and V-representation
msc: [52B11, 52B55, 90C05]
sources:
  - url: https://en.wikipedia.org/wiki/Convex_polytope
---

## Summary

A [[convex polytope|300K6CB1KH]] can be described in two ways.
A V-representation lists finitely many points whose [[convex hull|0FQSDC2EQ2]] is the polytope.
An H-representation lists finitely many linear inequalities, that is closed half-spaces, whose common solutions form the polytope.
Every convex polytope has both, and computations with polytopes, as in linear programming and computational geometry, depend on which one is at hand.

## Explanations

### L1 human

To describe a box you can give its eight corners, or say that each coordinate lies between two bounds.
The list of corners is a V-representation, and the list of bounds is an H-representation.
Some questions are easy with one description and hard with the other.
Checking that a point lies in the box is easy with the inequalities; finding the corner farthest in a given direction is easy with the corner list.