SL(n) Contravariant Matrix-Valued Valuations on Polytopes

TL;DR

This paper classifies all $ extrm{SL}(n)$ contravariant matrix-valued valuations on polytopes in $ eal^n$, removing symmetry and continuity assumptions, and identifies unique valuations in higher dimensions.

Contribution

It provides a complete classification of $ extrm{SL}(n)$ contravariant matrix-valued valuations on polytopes without continuity, including the removal of symmetry assumptions.

Findings

The Lutwak-Yang-Zhang matrix is the only valuation for $n extgreater=4$.

A new function appears in the classification for dimension three.

In dimension two, the classification aligns with known $ extrm{SL}(2)$ equivariant valuations.

Abstract

All $\textrm{SL}(n)$ contravariant matrix-valued valuations on polytopes in $\mathbb{R}^n$ are completely classified without any continuity assumptions. Moreover, the symmetry assumption of matrices is removed. The general Lutwak-Yang-Zhang matrix turns out to be the only such valuation if $n\geq 4$, while a new function shows up in dimension three. In dimension two, the classification corresponds to the known case of $\textrm{SL}(2)$ equivariant matrix-valued valuations.