The problem of finding minimal volume boxes circumscribing a given set of three-dimensional points is investigated. It is shown that it is not necessary for a minimum volume box to have any sides flush with a face of the convex hull of the set of points, which makes a naive search problematic. Nevertheless, it is proven that at least two adjacent box sides are flush with edges of the hull, and this characterization enables an O(n³) algorithm to find all minimal boxes for a set of n points.