The Feasible Set of a Linear Program is a Convex Polytope

Last updated Nov 1, 2022

Suppose $n, m \in \mathbb{N}$, $\mathbf{b}, \mathbf{c} \in \mathbb{R}^{n}$, $A \in \mathbb{R}^{m \times n}$ define the Linear Program $$\begin{align*} &&\max \mathbf{c}^{T} \mathbf{x}\\ &\text{s.t.}&A \mathbf{x} \leq \mathbf{b}\\ &&\mathbf{x} \geq 0 \end{align*}$$

Then the Feasible Set of this Linear Program is a Convex Polytope.

$A \mathbf{x} \leq \mathbf{b}$ is a system of Closed Halfspaces $\blacksquare$