SL($n$) contravariant tensor valuations of small orders

TL;DR

This paper classifies all 1SL(n)1 contravariant tensor valuations of small orders on convex polytopes, including symmetric and asymmetric types, and extends Minkowski relations to these valuations.

Contribution

It provides a complete classification of 1SL(n)1 contravariant tensor valuations without symmetry restrictions, introducing asymmetric variants linked to the cross tensor and Levi-Civita tensor.

Findings

Revealed asymmetric tensor valuations related to the cross tensor and Levi-Civita tensor.

Extended Minkowski relations to these asymmetric tensor valuations.

Unified classification of tensor valuations of small orders.

Abstract

A complete classification of \(\mathrm{SL}(n)\) contravariant, \(p\)-order tensor valuations on convex polytopes in \( \mathbb{R}^n \) for \( n \geq p \) is established without imposing additional assumptions, particularly omitting any symmetry requirements on the tensors. Beyond recovering known symmetric tensor valuations, our classification reveals asymmetric counterparts associated with the cross tensor and the Levi-Civita tensor. Additionally, some Minkowski type relations for these asymmetric tensor valuations are obtained, extending the classical Minkowski relation of surface area measures.