SL($n$) Contravariant Tensor Valuations of Orders up to $n$

Jin Li; Dan Ma

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arXiv · 2505.12277

SL($n$) Contravariant Tensor Valuations of Orders up to $n$

Jin Li·Dan Ma

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Abstract

A complete classification of $\mathrm{SL}(n)$ contravariant, $p$-order tensor valuations on 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 relations of surface area measures.

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Jin Li, Dan Ma. 2025-05-18. SL($n$) Contravariant Tensor Valuations of Orders up to $n$. https://arxiv.org/abs/2505.12277

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