  We study a polyhedron with $n$ vertices of fixed volume having minimum
surface area. Completing the proof of Fejes Toth, we show that all faces of a
minimum polyhedron are triangles, and further prove that a minimum polyhedron
does not allow deformation of a single vertex. We also present possible minimum
shapes for $n\le 12$, some of them are quite unexpected, in particular $n=8$.
