A set in $mathbb{R}^n$ is said to be polyhedral if it is the intersection of a finite number of closed half spaces, i.e.,

$S=left { x in mathbb{R}^n:p_{i}^{T}xleq alpha_i, i=1,2,….,n right }$

For example,

• $left { x in mathbb{R}^n:AX=b right }$

• $left { x in mathbb{R}^n:AXleq b right }$

• $left { x in mathbb{R}^n:AXgeq b right }$

Polyhedral Cone

A set in $mathbb{R}^n$ is said to be polyhedral cone if it is the intersection of a finite number of half spaces that contain the origin, i.e., $S=left { x in mathbb{R}^n:p_{i}^{T}xleq 0, i=1, 2,… right }$

Polytope

A polytope is a polyhedral set which is bounded.

Remarks

• A polytope is a convex hull of a finite set of points.
• A polyhedral cone is generated by a finite set of vectors.
• A polyhedral set is a closed set.
• A polyhedral set is a convex set.