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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 offaces and vertices. By distinguishing between various material distributionsand stability types, this expands into a small family of related questions.While many upper and lower bounds on the necessary numbers of faces andvertices have been established, none of these questions has been so farresolved. Adapting an algorithm presented in (Boz\'{o}ki et al., 2022), here weoffer the first complete answer to a question from this family: by using thetoolbox of semidefinite optimization to efficiently generate the hundreds ofthousands of infeasibility certificates, we provide the first-ever proof forthe existence of a monostatic polyhedron with point masses, having minimalnumber (V=11) of vertices (Theorem 3) and a minimal number (F=8) of faces. Wealso show that V=11 is the smallest number of vertices that a mono-unstablepolyhedron can have in all dimensions greater than 1. (Corollary 6.)

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