If $A$ and $B$ are polyhedra, how do we show that the intersection $A ∩ B$ is a polyhedron.
Does the same apply if they are both polytopes, will the intersection $A ∩ B$ also be a polytope?
The definition for a polyhedron is: A polyhedron $P ⊆ R^n$ is defined as the solution set of a system of linear inequalities. Thus, P has the form $P =$ {$x ∈ R^n: Ax ≤ b$}
Suppose that $P_1 = \{x:Ax\leq a\}$ and $P_2=\{x:Bx\leq b\}$ are the two polyhedron. Then $$ P_1\cap P_2 = \{x:Ax\leq a,\;Bx\leq b\} = \Bigg\{x : \begin{bmatrix}A\\B\end{bmatrix}x \leq \begin{bmatrix}a\\b\end{bmatrix}\Bigg\} = \{x:Cx\leq c\} $$ where $C = \begin{bmatrix}A\\B\end{bmatrix}$ and $c=\begin{bmatrix}a\\b\end{bmatrix}$. Thus $P_1\cap P_2$ can be represented as a polyhedron.