Facets of secondary polytopes and Chow stability

Chow stability is one of notions of Mumford's Geometric Invariant Theory to study the moduli space of polarized varieties. Kapranov, Sturmfels and Zelevinsky detected that Chow stability of polarized toric varieties is determined by its inherent {\it secondary polytope}, which is a polytope whose vertices are corresponding to regular triangulations of the associated (Delzant) polytope \cite{KSZ}. In this paper, we give a purely convex-geometrical proof that the Chow form of a smooth polarized toric variety is $H$-semistable if and only if it is $H$-polystable for the standard complex torus $H$.