I want to figure out which triangulations of the cube (i.e., partitions into tetrahedra using only the $8$ given vertices) are regular, but I'm not sure how to easily tell whether a given triangulation is regular. Is there some condition I can check relatively quickly to see if a particular triangulation is regular?

Edit: A triangulation of the cube is regular if there exists a convex piecewise-linear function from the cube to $\mathbb R$ such that the projection of the boundaries of the regions of linearity onto the cube gives the triangulation.