Lecture Notes 5 : Applications of Linear Programming

Proof. Let x ∈ P . We show a more general claim: if x tightly fulfills r independent constraints, then x can be expressed as a convex combination of at most n+ 1− r vertices. Substituting r = 0 we get the theorem. We prove the claim by induction on r, where the basis is r = n and we decrease r in each step. The basis of the induction is r = n and thus n constraints are fulfilled tightly. By theorem A, x is a vertex and thus can trivially be expressed as a combination of at most one vertex (itself). We assume that x tightly fulfills r independent constraints. Writing these constraints as equalities we get a smaller polytope P ′ ⊆ P such that x ∈ P ′. By theorem B we know that every polytope has a vertex, let v be an arbitrary vertex of P ′. Note that vertices of P ′ are also a vertices of P since a vertex is defined by any n tight constraints (according to theorem A), thus v is also a vertex of P . We now consider the ray starting at v and containing the segment that connects x and v: ∀t ≥ 0, φ(t) = v + t(x− v) Let t0 be the maximal t such that φ(t) is still feasible (Note that t0 > 1 since x ∈ P ′ and P ′ is convex). Denote y = φ(t0). Then y fulfills tightly the r original constraints plus one more, and by the inductive assumption it can be expressed as a convex combination of at most n+ 1− (r + 1) = n− r vertices of P . Denote this combination by