Lattice reduced and complete convex bodies

IF 1 2区数学 Q1 MATHEMATICS Journal of the London Mathematical Society-Second Series Pub Date : 2024-09-17 DOI:10.1112/jlms.12982

Giulia Codenotti, Ansgar Freyer

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Abstract

The purpose of this paper is to study convex bodies $C$ for which there exists no convex body $C^{'} ⊊ C$ of the same lattice width. Such bodies will be called ‘lattice reduced’, and they occur naturally in the study of the flatness constant in integer programming, as well as other problems related to lattice width. We show that any simplex that realizes the flatness constant must be lattice reduced and prove structural properties of general lattice reduced convex bodies: they are polytopes with at most $2^{d + 1} - 2$ vertices and their lattice width is attained by at least $Ω (\log d)$ independent directions. Strongly related to lattice reduced bodies are the ‘lattice complete bodies’, which are convex bodies $C$ for which there exists no $C^{'} ⊋ C$ such that $C^{'}$ has the same lattice diameter as $C$ . Similar structural results are obtained for lattice complete bodies. Moreover, various construction methods for lattice reduced and complete convex bodies are presented.

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晶格缩小和完整凸体

本文的目的是研究凸体 C $C$，对于这些凸体 C ′ ⊊ C $C^\prime \subsetneq C$，不存在网格宽度相同的凸体 C ′ ⊊ C $C^\prime \subsetneq C$。这样的体将被称为 "格子缩小体"，它们会自然地出现在整数编程中平坦常数的研究中，以及其他与格子宽度相关的问题中。我们证明了任何实现平整度常数的单纯形都必须是晶格缩小的，并证明了一般晶格缩小凸体的结构性质：它们是顶点至多为 2 d + 1 - 2 $2^{d+1}-2$ 的多面体，其晶格宽度至少由 Ω ( log d ) $\Omega (\log d)$ 独立方向达到。与晶格缩小体密切相关的是 "晶格完全体"，即不存在任何 C ′ ⊋ C $C^\prime \supsetneq C$ 使 C ′ $C^\prime$ 与 C $C$ 具有相同晶格直径的凸体 C $C$。类似的结构结果也适用于晶格完全体。此外，还提出了格子缩小凸体和完整凸体的各种构造方法。

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来源期刊

Journal of the London Mathematical Society-Second Series 数学-数学

CiteScore

1.90

自引率

0.00%

发文量

186

审稿时长

6-12 weeks

期刊介绍： The Journal of the London Mathematical Society has been publishing leading research in a broad range of mathematical subject areas since 1926. The Journal welcomes papers on subjects of general interest that represent a significant advance in mathematical knowledge, as well as submissions that are deemed to stimulate new interest and research activity.