Given: a set $S$ of points in $\mathbb{R}^3$.
Find: the smallest oriented bounding box that contains all the points. Note, the bounding box is "oriented" and thus need not be axis-aligned.
Can this be formulated as an optimization problem? Preferably a convex optimization problem!
The optimal box should be smallest by either volume or perimeter (whichever is easier, but ideally, both solutions will be presented).
I hope the optimization problem is simpler to code than Joseph O'Rourke's Minimum Enclosing Box Algorithm (pdf) and can be solved using a standard optimization package.
For example, the smallest enclosing hypersphere containing a set of points $S$ can be defined by a center $c$ and radius $r$. It is the solution to the problem:
$\textrm{minimize}_{c,r} \; r^2 \;\;$ $ \textrm{subject to: } \; \left| \left| c - s \right| \right|^2 \leq r^2 \; \forall s \in S$
I'm asking for a similar formulation for an oriented bounding box instead of a hypersphere, if possible.