Let $M$ be a centered parallelepiped, the intersection of $M$ and any plane $P$ that passes through the origin is a parallelogram or hexagon. Each parallelogram or hexagon has a cubic box that is the smallest box that can contain the parallelogram or hexagon. Denote the cubic box by $B(P)$. There exist planes $P_{0}$ such that $B(P_{0})$ is the smallest among all the boxes $B(P)$.

Is it true that there is always one of the planes $P_{0}$ such that the cross-section of the centered parallelepiped $M$ by $P_{0}$ is a parallelogram?

Thanks.

adjacentedges, where adjacent means they are incident to a common vertex. Or: only intersects the interiors of nonadjacent edges. In any case, I do not see an immediate counterexample... – Joseph O'Rourke Nov 28 '10 at 0:54