A theory of polyhedra
What has been called by \textit{J. Lawrence} [Discrete Comput. Geom. 3, No. 4, 307-324 (1988; Zbl 0646.52003)] (and unpublished notes) and \textit{K. G. Fischer} and \textit{J. Shapiro} [J. Pure Appl. Algebra 78, No. 3, 239-251 (1992; Zbl 0756.52015)] the Minkowski ring is the additive group with a generator \([P]\) for each convex polytope \(P\) in \(R^ d\), satisfying the relations \([P \cup Q] + [P\cap Q] = [P] + [Q]\) whenever \(P \cup Q\) is also convex (corresponding to the valuation property; \([\emptyset] = 0\)), with multiplication induced by Minkowski addition, so that \([P]\cdot [Q] = [P + Q]\). Here author works with the not necessarily bounded convex polyhedra in a module over an ordered ring, initially showing that the earlier theory generalizes in a natural way. A central result is that the mapping taking a polyhedron into the exponential of its support functional induces a separating ring homomorphism on the Minkowski ring. When translation invariance \([P + t] = [P]\) for translation vectors \(t\) is additionally imposed, in \(R^ d\) one obtains the polytope algebra of the reviewer [Adv. Math. 78, No. 1, 76-130 (1989; Zbl 0686.52005)]. The author reproves many of the results of the latter paper in the more general context. The most novel part of the paper is the last. There the author imposes a \(\lambda\)-ring structure on the ring; this is developed in various ways, generalizing notions from \(K\)-theory. Results of note here describe the lattice point enumerator in terms of characteristic classes, and give a combinatorial Riemann-Roch theorem. The paper provides a foundation for valuation theory on general polyhedra, and insights into possible future directions for research.
scientific article; zbMATH DE number 221225