Decomposing the secondary Cayley polytope

The vertices of the secondary polytope of a point configuration correspond to its regular triangulations. The Cayley trick links triangulations of one point configuration, called the Cayley polytope, to the fine mixed subdivi sions of a tuple of point configurations. In this paper we investigate the secondary polytope of this Cayley polytope. Its vertices correspond to all regular mixed subdivisions of a tuple of point configurations. We demonstra te that it equals the Minkowski sum of polytopes, which we call mixed secon dary polytopes, whose vertices correspond to regular-cell configurations.