Is there always a parallelogram cross-section of parallelepiped contained in the smallest box Dear Sergei, your example is very interesting. If you plot the image of $Q$ under $A$, $A(Q)$ is not a parallelepiped, as it has more than 6 faces. The question should be: If $A(Q)$ is a parallelepiped, is there always one of the planes $P_0$ such that the plane does not intersect with the interior of any two adjacent edges?