TITLE: How to check whether a convex polyhedron is contained in another convex polyhedron?
QUESTION [6 upvotes]: Suppose we have two convex polyhedra 
$$P_1=\{x\in \mathbb{R}^n \mid A_1 x \geq b_1 \}$$
$$P_2=\{x\in \mathbb{R}^n \mid A_2 x \geq b_2 \}$$
Is there a way to check whether $P_1 \subseteq P_2$? I was hoping that this can be done by solving some linear program. Is this possible?

REPLY [2 votes]: This is equivalent to the feasibility of the following linear constraints:
$$
\Lambda A_2 = A_1, \Lambda b_2 \ge b_1,
$$ 
where $\Lambda$ is a matrix with non-negative entries. The proof follows from the dual of linear programs that Michael mentioned.
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