Valuations on polyhedra and topological arrangements

TL;DR

This paper explores a generalized framework for valuations on convex polyhedra, linking scissors congruence problems with hyperplane arrangements and extending to topological arrangements with non-straight hyperplanes.

Contribution

It introduces a broad setting for valuations on polyhedra based on hyperplane collections without algebraic restrictions, revealing new connections with hyperplane arrangements and topological generalizations.

Findings

Established a relationship between valuations and hyperplane arrangements.

Extended the theory to topological arrangements with nontrivial topology.

Connected properties of Varchenko--Gelfand algebras to polytope rings.

Abstract

We revisit a classical theme of (general or translation invariant) valuations on convex polyhedra. Our setting generalizes the classical one, in a ``dual'' direction to previously considered generalizations: while previous research was mostly concerned with variations of ground fields/rings, over which the vertices of polytopes are defined, we consider more general collections of defining hyperplanes. No algebraic structures are imposed on these collections. This setting allows us to uncover a close relationship between scissors congruence problems on the one hand and finite hyperplane arrangements on the other hand: there are many parallel results in these fields, for which the parallelism seems to have gone unnoticed. In particular, certain properties of the Varchenko--Gelfand algebras for arrangements translate to properties of polytope rings or valuations. Studying these properties is possible in a general topological setting, that is, in the context of the so-called topological arrangements, where hyperplanes do not have to be straight and may even have nontrivial topology.