Scissors Congruence with Mixed Dimensions

TL;DR

This paper introduces new algebraic groups for bounded polytopes in Euclidean space that incorporate lower-dimensional parts and relate to spherical scissors congruence, generalizing the Dehn invariant.

Contribution

It defines the Grothendieck group $E_n$ for bounded polytopes including lower dimensions and introduces $L_n$ related to spherical scissors congruence, extending classical concepts.

Findings

Defines the group $E_n$ for bounded polytopes with lower-dimensional parts.

Introduces the group $L_n$ using germs of polytopes, connected to spherical scissors congruence.

Provides a framework for a generalized Dehn invariant.

Abstract

We introduce a Grothendieck group $E_n$ for bounded polytopes in $\mathbb R^n$. It differs from the usual Euclidean scissors congruence group in that lower-dimensional polytopes are not ignored. We also define an analogous group $L_n$ using germs of polytopes at a point, which is related to spherical scissors congruence. This provides a setting for a generalization of the Dehn invariant.