Main Theorem for Polytopes (1.1)
A subset \(P\subseteq\mathbb{R}^d\) is the convex hull of a finite point set (a \(V\)-polytope)
\begin{align*}
P=\operatorname{conv}(V) \quad \quad \text{ for some } V\in\mathbb{R}^{d\times n}
\end{align*}
if and only if it is a bounded intersection of halfspaces (an \(H\)-polytope)
\begin{align*}
P=P(A,z) \quad \quad \text{ for some } A\in\mathbb{R}^{m\times d}, z\in\mathbb{R}^m
\end{align*}
This the version we need since we’re studying polytopes but the version we will prove is the following where remove the “bounded” constraint and get
Main Theorem for Polyhedra (1.2)
A subset \(P\subseteq\mathbb{R}^d\) is a sum of a convex hull of a finite set of points plus a conical combination of vectors (a \(V\)-polyhedron)
\begin{align*}
P=\operatorname{conv}(V)+\operatorname{cone}(Y)
\quad \quad \text{ for some }
V\in\mathbb{R}^{d\times n}, Y\in\mathbb{R}^{d\times n'}
\end{align*}
if and only if it is an intersection of closed halfspaces (an \(H\)-polyhedron)
\begin{align*}
P=P(A,z)
\quad \quad \text{ for some }
A\in\mathbb{R}^{m\times d},\; z\in\mathbb{R}^m
\end{align*}
References
- Wikipedia
- Lectures on Polytopes