Secondary power diagram, dual of secondary polytope

An ingenious construction of Gel’fand et al. (Discriminants, Resultants, and Multidimensional Determinants. Birkhäuser, Basel, 1994) geometrizes the triangulations of a point configuration, such that all coherent triangulations form a convex polytope, the so-called secondary polytope. The secondary polytope can be treated as a weighted Delaunay triangulation in the space of all possible coherent triangulations. Naturally, it should have a dual diagram. In this work, we explicitly construct the secondary power diagram, which is the power diagram of the space of all possible power diagrams with non-empty boundary cells. Secondary power diagram gives an alternative proof for the classical secondary polytope theorem based on Alexandrov theorem. Furthermore, secondary power diagram theory shows one can transform a non-degenerated coherent triangulation to another non-degenerated coherent triangulation by a sequence of bistellar modifications, such that all the intermediate triangulations are non-degenerated and coherent.