Beitraege zur Algebra und Geometrie
Contributions to Algebra and Geometry
Volume 34 (1993), No. 1, 15-30.
Separation in the Polytope Algebra
Peter McMullen
Abstract.
The polytope algebra is the universal group for translation invariant
valuations on the family of polytopes in a finite dimensional vector space
over an ordered field. In an earlier paper, it was shown that the polytope
algebra is, in all but one trivial respect, a graded (commutative) algebra
over the base field. Also described was a family of separating (group)
homomorphisms, called frame functionals. However, various questions relating
to the frame functionals were left open, such as what syzygies exist between
them, and what the image of a certain closely related mapping is. Here, these
questions are settled: essentially, the only restrictions are imposed by the
Minkowski relations. In doing this, simpler proofs are also found of some
results in that earlier paper. Finally, there are consequences for expressing
certain translation invariant valuations in terms of mixed volumes.
MSC 1991: Primary 52B45; dissections and valuations.