Question

prove that polyhedron is a convex set​

Answer 1

It can be proved by following three steps.

(a) Let {Ωα}(α∈I) be a collection of convex subsets of Rn. Then ⋂α∈IΩα is also a convex.

proof: Taking any x1,x2∈⋂α∈IΩα. We get that x1,x2∈Ωα for α∈I. And then we have θx1+(1−θ)x2∈Ωα for any θ∈[0,1] since Ωα are convex sets. Thus θx1+(1−θ)x2∈⋂α∈IΩα.

(b) Hyperplanes are convex and halfsapces are also convex.

Hyperplanes:{x|aTx=b}Halfspaces:{x|aTx≤b}

proof: Assume that x1,x2∈Ω, and we have aTx1=b,aTx2=b. Hence we can get

aT(θx1+(1−θ)x2)=θaTx1+(1−θ)aTx2=b

i.e., (θx1+(1−θ)x2)∈Ω. similarly, we also can prove that halfspaces are convex.

(c) As we observed from the definition of polyhedra. A polyhedron is defined as the solution set of a finite number of linear equalities and inequalities. It mean that a ployhedron is the intersection of a finite number of halfspaces and hyperplanes. Based on (b), we know that halfspaces and hyperplanes are convex.

Hope it helps