A Valuation-Theoretic Approach to Translative-Equidecomposability

TL;DR

This paper classifies all simple translation-invariant valuations on polytopes, providing new insights into translative-equidecomposability conditions and offering a simplified proof for the classification of continuous valuations.

Contribution

It introduces a valuation-theoretic framework that unifies and simplifies the classification of translation-invariant valuations on polytopes.

Findings

Complete classification of simple translation-invariant valuations

Recovery of known conditions for translative-equidecomposability

Simplified proof of classification for continuous valuations

Abstract

All simple translation-invariant valuations on polytopes are classified. As a direct consequence the well-known conditions for translative-equidecomposability are recovered. Furthermore, a simplified proof of the classification of continuous simple translation-invariant valuations is presented.