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

TL;DR

This paper proves the existence of the smallest mono-unstable convex polyhedron with point masses, having 8 faces and 11 vertices, using semidefinite optimization to resolve a longstanding question in polyhedral stability.

Contribution

It provides the first complete proof of the minimal face and vertex counts for mono-unstable convex polyhedra with point masses, expanding understanding of monostatic polyhedra.

Findings

Existence of a convex mono-unstable polyhedron with 8 faces and 11 vertices.

11 vertices is the minimal number for mono-unstable polyhedra in all dimensions greater than 1.

Semidefinite optimization effectively generates infeasibility certificates for polyhedral stability analysis.

Abstract

In the study of monostatic polyhedra, initiated by John H. Conway in 1966, the main question is to construct 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 (Boz\'{o}ki et al., 2022), here we offer the first complete answer to a question from this family: by using the toolbox of semidefinite 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 number (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.)