# Introduction

Let $\mathbb{F}$ be an ordered field; the default is that $\mathbb{F}=\mathbb{Q}$ is the field of rational numbers and other fields are not yet supported everywhere in the implementation.

A set $P \subseteq \mathbb{F}^n$ is called a (convex) polyhedron if it can be written as the intersection of finitely many closed affine halfspaces in $\mathbb{F}^n$. That is, there exists a matrix $A$ and a vector $b$ such that $P = P(A,b) = \{ x \in \mathbb{F}^n \mid Ax \leq b\}.$ Writing $P$ as above is called an $H$-representation of $P$.

When a polyhedron $P \subset \mathbb{F}^n$ is bounded, it is called a polytope and the fundamental theorem of polytopes states that it may be written as the convex hull of finitely many points. That is $P = \textrm{conv}(p_1,\ldots,p_N), p_i \in \mathbb{F}^n.$ Writing $P$ in this way is called a $V$-representation. Polytopes are necessarily compact, i.e., they form convex bodies.

Each polytope has a unique $V$-representation which is minimal with respect to inclusion (or cardinality). Conversely, a polyhedron which is full-dimensional, has a unique minimal $H$-representation. If the polyhedron is not full-dimensional, then there is no canonical choice of an $H$-representation.