Polyhedra vs Polytope

Question

I am having a hard time understanding what is the main difference between a polyhedron and a polytope. Could anyone explain me what is the difference between these two structures?

Answer 1

A polyhedron is a special case of a polytope, or, equivalently, a polytope is a generalization of a polyhedron. A polytope has a certain dimension $n$, and when $n=3$ we say that the polytope is a polyhedron. (Similarly when $n=2$ we say that the polytope is a polygon.)

This is analogous to how we can define a general $n$-dimensional sphere, and how we call the $n=1$ case a "circle".

EDIT: Indeed I should mention that this definition is not universal. Some people say "polyhedron" to mean "polytope" as I've used it above, and say "polytope" to mean "bounded polyhedron".

Answer 2

According to Section 3.5 of the book Integer Programming by Conforti, Cornuejols and Zambelli, a subset $P\in \mathbb{R}^n$ is a polyhedron if there exists a positive integer $m$, an $m\times n$ matrix $A$ and a vector $b\in\mathbb{R}^m$ such that $$ P=\{x\in \mathbb{R}^n: Ax\le b\} $$ For polytopes, the book says a subset $Q$ is a polytope if $Q$ is the convex hull of a finite set of vectors in $\mathbb{R}^n$.

To understand their relation, I think the Minkowski-Weyl Theorem makes a very clear claim. But before the Theorem, we need another definition called polyhedral cone (or finitely generated cone, these terms are equivalent) which is defined as the intersection of a finite number of half-spaces containing the origin on the boundaries, that is $$ C := \{x\in\mathbb{R}^n:Ax\le 0\} $$ With these definitions,

(Minkowski-Weyl Theorem, Theorem 3.13 of the CCZ book) A subset $P$ of $\mathbb{R}^n$ is a polyhedron if and only if $P= Q+C$ for some polytope $Q\subset \mathbb{R}^n$ and finitely generated cone $C\subseteq \mathbb{R}^n$.