Question

I have an optimization question if anyone's up for it, thanks\

13. Prove that the intersection of a finite number of polyhedra in R" is a polyhedron in RT

Answer 1

A polyhedron is the intersection of finitely many halfspaces. So let us consider finitely many polyhedra in

$\mathbb{R}^n$ as

P.= n tj. € R" : 1), 4.. 2 bis} ji=1 for i=1,2, ..., k.

Then the intersection of these polyhedra is

k mi DENNE, EX": 0,4, 2 0.1 - (non ER":1, 2 bu?) n (ne; ER" : olyan 20x8) --( , ER":, 1, 2 buk) i=1 i=1j=1 Vi=1 12=1 k=1

which is the intersection of finitely many halfspaces and is thus a polyhedron in

$\mathbb{R}^n$.

Hence proved.