# Constraint Set of Canonical Linear Programming Problem is Convex

I'm reading through my first textbook on linear optimization. The book states a theorem without proof and I'd like to understand why it's true.

Glossary of Terms:

Definition 1

The problem

Maximize $f(x_1,x_2,\cdots,x_n)=c_1x_1+c_2x_2+\cdots+c_nx_n$

Subject to $a_{11}x_1+a_{12}x_2+\cdots+a_{1n}x_n\leq b_1$

$a_{21}x_1+a_{22}x_2+\cdots+a_{2n}x_n \leq b_2$

$\cdots$

$a_{m1}x_1+a_{m2}x_2+ \cdots + a_{mn}x_n \leq b_n$

$x_1,x_2, \dots, x_n \geq 0$

is said to be a canonical maximization linear programming problem. The definition for minimization is analogous.

Definition 2

Let $x= (x_1,x_2,\cdots ,x_n), y=(y_1,y_2,\cdots ,y_n)\in$ R$^n$. Then $tx+(1-t)y$ for $0\leq t\leq 1$ is said to be the line segment between $x$ and $y$ inclusive.

Definition 3

The set of all points $(x_1,x_2, \cdots, x_n)$ satisfying the constraints of the canonical maximization problem is said to be the constraint set

Definition 4