What is the practical purpose you intend to use the for? That would help in knowing what information you have available. Looks like it'll be tough though.
In 3D, it is considerably more complicated to find the minimum volume bounding box of a convex polyhedron. The current fastest algorithm is by [O'Rourke, 1985], and it runs in O(n3) time. But it may be possible to improve on this. Regardless, since O(n3) algorithms are slow in practice for large point sets, there is considerable interest in fast approximations for the minimum volume box in 3D and higher dimensions. Recently, [Barequet & Har-Peled, 1999] have described two randomized algorithms that run in O(n log n + 1/å4.5) and O(n log n + n/å3) expected time, where Ä is the error of the approximation. The deterministic variants of these algorithms run in O(n log2 n + 1/å4.5) and O(n log2 n + n/å3) time. Only the second of these algorithms is simple enough to implement in practice.
In 3D, it is considerably more complicated to find the minimum volume bounding box of a convex polyhedron. The current fastest algorithm is by [O'Rourke, 1985], and it runs in O(n3) time. But it may be possible to improve on this. Regardless, since O(n3) algorithms are slow in practice for large point sets, there is considerable interest in fast approximations for the minimum volume box in 3D and higher dimensions. Recently, [Barequet & Har-Peled, 1999] have described two randomized algorithms that run in O(n log n + 1/å4.5) and O(n log n + n/å3) expected time, where Ä is the error of the approximation. The deterministic variants of these algorithms run in O(n log2 n + 1/å4.5) and O(n log2 n + n/å3) time. Only the second of these algorithms is simple enough to implement in practice.