Abstract We show that the Grothendieck group associated to integral polytopes in ℝn is free-abelian, by providing an explicit basis. Moreover, we identify the involution on this polytope group given by reflection about the origin as a sum of Euler characteristic type. We also compute the kernel of the norm map sending a polytope to its induced seminorm on the dual of ℝn.

中文翻译：

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积分多面体群

摘要 我们通过提供一个明确的基础证明了与ℝn 中的积分多胞体相关的格罗腾迪克群是自由阿贝尔群。此外，我们将通过关于起源的反思给出的这个多胞体群的对合识别为欧拉特征类型的总和。我们还计算了范数映射的内核，将多面体发送到其在 ℝn 的对偶上的诱导半范数。