Comparison of convex hulls and box hulls

A convex hull of a set of points X is the minimal convex set containing X. A box B is an interval B = {x|x ∈ [a, b], a, b ∈ Rⁿ}. A box hull of a set of points X is defined to be the minimal box containing X. Because both convex hulls and box hulls are closure operations of points, classical results for convex sets can naturally be extended for box hulls. We consider here the extensions of theorems by Carathéodory, Helly and Radon to box hulls and obtain exact results.