Polyhedral Gauss-Bonnet theorems and valuations

TL;DR

This paper generalizes the Gauss-Bonnet theorem for polyhedra in Euclidean space by introducing valuation-based curvatures, leading to new polyhedral Gauss-Bonnet theorems.

Contribution

It proposes a valuation-based framework for defining vertex curvatures, extending classical polyhedral Gauss-Bonnet theorems to a broader setting.

Findings

Introduces valuation-based vertex curvatures

Derives new polyhedral Gauss-Bonnet theorems

Provides a unifying framework for curvature and topology

Abstract

The Gauss-Bonnet theorem for a polyhedron (a union of finitely many compact convex polytopes) in $n$-dimensional Euclidean space expresses the Euler characteristic of the polyhedron as a sum of certain curvatures, which are different from zero only at the vertices of the polyhedron. This note suggests a generalization of these polyhedral vertex curvatures, based on valuations, and thus obtains a variety of polyhedral Gauss-Bonnet theorems.