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arXiv · 2405.06897

SL(n) Contravariant Matrix-Valued Valuations on Polytopes

Chunna Zeng·Yuqi Zhou
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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.

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Chunna Zeng, Yuqi Zhou. 2024-05-11. SL(n) Contravariant Matrix-Valued Valuations on Polytopes. https://arxiv.org/abs/2405.06897

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