  I prove the existence, and describe the structure, of moduli space of pairs
$(p,\Theta)$ consisting of a projective variety $P$ with semiabelian group
action and an ample Cartier divisor on it satisfying a few simple conditions.
Every connected component of this moduli space is proper. A component
containing a projective toric variety is described by a configuration of
several polytopes, the main one of which is the secondary polytope. On the
other hand, the component containing a principally polarized abelian variety
provides a moduli compactification of $A_g$. The main irreducible component of
this compactification is described by an "infinite periodic" analog of the
secondary polytope and coincides with the toroidal compactification of $A_g$
for the second Voronoi decomposition.
