The smallest mono-unstable convex polyhedron with point masses has 8 faces and 11 vertices

In the study of monostatic polyhedra, initiated by John H. Conway in 1966, the main question is to con-struct such an object with the minimal number of faces and vertices. By distinguishing between various material distributions and stability types, this expands into a small family of related questions. While many upper and lower bounds on the necessary numbers of faces and vertices have been established, none of these questions has been so far resolved. Adapting an algorithm presented in Bozoki et al. (2022), here we offer the first complete answer to a question from this family: by using the toolbox of semidefi-nite optimization to efficiently generate the hundreds of thousands of infeasibility certificates, we provide the first-ever proof for the existence of a monostatic polyhedron with point masses, having minimal num-ber ( V = 11 ) of vertices (Theorem 3) and a minimal number ( F = 8 ) of faces. We also show that V = 11 is the smallest number of vertices that a mono-unstable polyhedron can have in all dimensions greater than 1 (Corollary 6). & COPY; 2023 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license ( http://creativecommons.org/licenses/by-nc-nd/4.0/ )

Automated summary

In the study of monostatic polyhedra, the main question is to construct an object with the minimal number of faces and vertices. Despite establishing upper and lower bounds on the necessary numbers, none of the questions have been resolved.