Approximating the Directional Width of a Shape

x " From the days of John the Baptist until now, the kingdom of heaven suffereth violence, and the violent bear it away. " – – Matthew 11:12 denote the directional width of P in the direction of v. A set Q ⊆ P is a ε-coreset for directional width, if ∀v ∈ S (d−1) ω(v, Q) ≥ (1 − ε)ω(v, P). Namely, the coreset Q provides a concise approximation to the directional width of P. The usefulness of such a coreset might become clearer in the light of the following claim. Claim 20.1.1 Let P be a set of points in IR d , 0 < ε ≤ 1 a parameter and let Q be a δ-coreset of P for directional width, for δ = ε/(8d). Let B opt (Q) denote the minimum volume bounding box of Q. Let B be the rescaling of B opt (Q) around its center by a factor of (1 + 3δ). Then, P ⊂ B , and in particular, Vol(B) ≤ (1 + ε)B opt (P), where B opt (P) denotes the minimum volume bounding box of P.