All \(\mathrm{SL}(n)\) 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 class of such valuations for \(n\ge 3\). In dimension two, the classification corresponds to the known case of \(SL (2)\) covariant valuations.

中文翻译：

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SL(n) 逆变向量估值

\({\mathbb {R}}^n\)中多胞体的所有\(\mathrm{SL}(n)\)逆变向量估值都被完全分类，没有任何额外的假设。定义了刻面向量。事实证明，对于\(n\ge 3\) 来说，这种估值是独一无二的。在维度二中，分类对应于\(SL (2)\)协变估值的已知情况。