A computational approach to stability problems

A polytope in R d is said to be stable on a facet if and only if the perpendicular to that facet through the center of gravity meets the facet itself. One may consider the stability of a body with respect to an arbitrary "center of gravity on the interior of the body [1,3,6], or one may take the center of gravity to be determined by the shape of the body itself. Most typically, the center of gravity is taken to be that point which would be the physical center of gravity were the body composed of a material of uniform density[1,2,4,5,6,7]; however, other possibilities, such as the center of mass determined by a uniform distribution on the nskeleton of the body for 1≤n≤d-1 may also be considered [5]. This type of problem seems to have been first introduced in two papers [1,7] that appeared in SIAM Review in the late 1960’s. In the second of these, the question of existence of polytopes stable only on one facet was raised. A 19-faceted polyhedron, stable only on one facet. was exhibited; and the construction was modified to yield a 23-faceted polytope stable on only its smallest facet. It was shown that no tetrahedron of uniform density could be monostatic; and it was asked whether any simplex of uniform density could have this property