Title: Minimum polyhedron with $n$ vertices,
Abstract: We study a polyhedron with $n$ vertices of fixed volume having minimum
surface area. Completing the proof of 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 $n=8$.