Advanced Geometric Methods in Computer Science

Problem B1 (20 pts). A ray , R, of E is any subset of the form R = {a+ tu | t ∈ R, t ≥ 0}, for some point, a ∈ E and some nonzero vector, u ∈ R. A subset, A ⊆ E, is unbounded iff it is not contained in any ball. Prove that every closed and unbounded convex set, A ⊆ E, contains a ray. Hint : Pick some point, c ∈ A, as the origin and consider the sphere, S(c, r) ⊆ E, of center c and radius r > 0. As A is unbounded and c ∈ A, show that Ar = A ∩ S(c, r) 6= ∅ for all r > 0. As A is closed, each Ar is compact. Consider the radial projection, πr : S(c, r)→ S(c, 1), from S(c, r) onto the unit sphere, S(c, 1), of center c given by