  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.)
