# polyhedra and extreme points

I am stuck with solving this problem, does anybody has idea, how to solve it ?

Let $P$ and $Q$ be polyhedra in $\mathbb{R}^n$. Let $P +Q := \{x+y ~\vert~ x \in P; y \in Q \}$

a) Show that $P + Q$ is a polyhedron.

b) Show that every extreme point of $P + Q$ is the sum of an extreme point of $P$ and an extreme point of $Q$.

• Can you add your definition of a polyhedron and what have you tried? Also, is this homework? – Kris Williams May 11 '13 at 13:35
• The second part of your question is easily settled : if $x$ is the midpoint of a nontrivial segment $[a,b]$ in $P$, then so is $x+y$ for any $y\in Q$ ($[a+y,b+y]$). For the first part, you should tell us your definition of polyhedron, as already suggested by @KrisWilliams. – Olivier Bégassat May 11 '13 at 14:07
• Thank you Olivier. intersection of some linear constraints create polydra and represented by Ax = b , where X > = 0 and dimention of A is m*n. – user51661 May 11 '13 at 19:03