Delaunay decompositions in dimension 4

The Voronoi cone decompositions has been attracting our attention in the compactification problem of the moduli scheme of abelian varieties. The objects to add as the boundary of the moduli scheme are stable quasi-abelian schemes, reduced or nonreduced, which are described in terms of Delaunay decompositions. In this note, we prove that any Delaunay decomposition is simplicially generating if the dimension is less than 5. As a corollary, if the dimension is less than 5, projectively stable quasi-abelian schemes are reduced, and therefore two kinds of stable quasi-abelian schemes, projectively stable quasi-abelian schemes and torically stable quasi-abelian varieties, are the same class of varieties.