A brief introduction to valuations on lattice polytopes
A brief introduction to valuations on lattice polytopes
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格子多面体估值简介
DOI：
10.1142/9789811200489_0002
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发表时间：
2019
期刊：
影响因子：
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通讯作者：
Katharina Jochemko
中科院分区：
文献类型：
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作者：
Katharina Jochemko
这些笔记是基于作者在2018年大坂大学的“晶格多面体夏季研讨会”上提供的五堂暑期学校课程。本文简要介绍了格多面体上的赋值理论。赋值是凸几何中的一个经典问题。体积起着重要的作用，在许多结构的结果，如哈德维格的著名的表征连续，刚性运动不变的评价凸机构。定义域仅限于格多面体的赋值研究较少。Betke-Kneser定理为格多面体上的格不变赋值建立了一个有趣的离散Hadwiger定理，其中格点的数量-离散体积-起着重要作用。从那里，我们探索凸体上的估值世界与格多面体上的估值世界之间惊人的相似之处，类比和差异，重点是积极性问题和与Ehrhart理论的联系。
These notes are based on a five-lecture summer school course given by the author at the “Summer Workshop on Lattice Polytopes” at Osaka University in 2018. We give a short introduction to the theory of valuations on lattice polytopes. Valuations are a classical topic in convex geometry. The volume plays an important role in many structural results, such as Hadwiger’s famous characterization of continuous, rigid-motion invariant valuations on convex bodies. Valuations whose domain is restricted to lattice polytopes are less well-studied. The Betke-Kneser Theorem establishes a fascinating discrete analog of Hadwiger’s Theorem for lattice-invariant valuations on lattice polytopes in which the number of lattice points — the discrete volume — plays a fundamental role. From there, we explore striking parallels, analogies and also differences between the world of valuations on convex bodies and those on lattice polytopes with a focus on positivity questions and links to Ehrhart theory.