SL($n$) contravariant vector valuations

TL;DR

This paper classifies all SL(n) contravariant vector valuations on polytopes in R^n, identifying a unique valuation in higher dimensions and relating it to the 2D case, advancing the understanding of valuation symmetries.

Contribution

It provides a complete classification of SL(n) contravariant vector valuations on polytopes, revealing the facet vector as the unique valuation for n ≥ 3 and connecting to SL(2) covariant valuations.

Findings

Facet vector is the unique valuation for n ≥ 3

Complete classification of SL(n) contravariant vector valuations

Connection between 2D covariant and higher-dimensional contravariant valuations

Abstract

All SL($n$) contravariant vector valuations on polytopes in $\mathbb R^n$ are completely classified without any additional assumptions. The facet vector is defined. It turns out to be the unique such valuation for $n\geq3$. In dimension two, the classification corresponds to the case of SL(2) covariant valuations.