Math Problem Solver

Let P ⊆ R

n be a nonempty polyhedron given by

P =







x ∈ R

n

:

(a

i

)

T x ≥ bi

, i ∈ M1,

(a

i

)

T x ≤ bi

, i ∈ M2,

(a

i

)

T x = bi

, i ∈ M3







,

where M1, M2, and M3 are finite index sets, a

i ∈ R

n and bi ∈ R for each i ∈ M1 ∪ M2 ∪ M3.

Prove the following result:

P ⊆ R

n has no vertices if and only if span{a

i

: i ∈ M1 ∪ M2 ∪ M3} ̸= R

n.