Reference no: EM131085693

Math 121c: Topics in Geometric Combinatorics, Spring 2012 Problems-

A set S ⊂ Rⁿ is a convex cone if it is a convex set, and for any x ∈ S and λ ≥ 0, λ_x ∈ S. Let S ⊂ Rⁿ. The convex cone generated by S, written coco(S), is the smallest convex cone containing S.

(a) Draw a few pictures to get used to coco(S).

(b) Prove that coco(S) = {_i=1∑^mλ_ix⁽ⁱ⁾: m ∈ N, λ_i ≥ 0, x⁽ⁱ⁾ ∈ S for i ∈ {1, 2, . . . , m}}.

(c) Adapt the proof of Caratheodory's Theorem to show that if S ⊂ Rⁿ, any point x ∈ coco(S) can be written as _i=1∑^m λ_ix⁽ⁱ⁾ with λ_i ≥ 0, x⁽ⁱ⁾ ∈ S for all i, and {x⁽¹⁾, x⁽²⁾, . . . , x^(m)} linearly independent.

(d) If S is a compact convex set must coco(S) be closed (in the usual topology on Rⁿ)? If so, why? If not, is the statement true when S is finite?