Given a convex set of $N$ points ($N\;\ge\;2$) in general positions. I want to solve the task of finding arbitrary oriented minimum volume cube, which contain all the points, and at least one point should lie on each facet of the cube. Say, for set of 2 points both of them should be vertices of the sought-for cube lies on the ends of its main diagonal.
Let $(\mathbf e_a, \mathbf e_b, \mathbf e_c)$ are orthogonal basis and all the basis vectors are codirectional to cube edges. $|\mathbf e_a| \overset{def}{\equiv} |\mathbf e_b| \overset{def}{\equiv} |\mathbf e_c| = A$. Need to minimize $A$.
The task can be reformulated: need to find such a basis (and an origin), that coordinates of all the points of input set should be in $[0;1]$, each coordinate should take on a values $0$ and $1$ at least once and unit vector of such a basis should have minimal length in source basis. Surely such a basis is not unique.
Can this task be solved analytically (or with minimal enumeration/search)? Say, for simplest case of a convex set of $N = 6$ points in general positions (I think this is a simplest non-trivial case).
