Lattice reduced and complete convex bodies

IF 16.4 1区化学 Q1 CHEMISTRY, MULTIDISCIPLINARY

Accounts of Chemical Research Pub Date : 2024-09-17 DOI:10.1112/jlms.12982

Giulia Codenotti, Ansgar Freyer

{"title":"Lattice reduced and complete convex bodies","authors":"Giulia Codenotti, Ansgar Freyer","doi":"10.1112/jlms.12982","DOIUrl":null,"url":null,"abstract":"The purpose of this paper is to study convex bodies <math>\n <semantics>\n <mi>C</mi>\n <annotation>$C$</annotation>\n </semantics></math> for which there exists no convex body <math>\n <semantics>\n <mrow>\n <msup>\n <mi>C</mi>\n <mo>′</mo>\n </msup>\n <mi>⊊</mi>\n <mi>C</mi>\n </mrow>\n <annotation>$C^\\prime \\subsetneq C$</annotation>\n </semantics></math> 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 <math>\n <semantics>\n <mrow>\n <msup>\n <mn>2</mn>\n <mrow>\n <mi>d</mi>\n <mo>+</mo>\n <mn>1</mn>\n </mrow>\n </msup>\n <mo>−</mo>\n <mn>2</mn>\n </mrow>\n <annotation>$2^{d+1}-2$</annotation>\n </semantics></math> vertices and their lattice width is attained by at least <math>\n <semantics>\n <mrow>\n <mi>Ω</mi>\n <mo>(</mo>\n <mi>log</mi>\n <mi>d</mi>\n <mo>)</mo>\n </mrow>\n <annotation>$\\Omega (\\log d)$</annotation>\n </semantics></math> independent directions. Strongly related to lattice reduced bodies are the ‘lattice complete bodies’, which are convex bodies <math>\n <semantics>\n <mi>C</mi>\n <annotation>$C$</annotation>\n </semantics></math> for which there exists no <math>\n <semantics>\n <mrow>\n <msup>\n <mi>C</mi>\n <mo>′</mo>\n </msup>\n <mo>⊋</mo>\n <mi>C</mi>\n </mrow>\n <annotation>$C^\\prime \\supsetneq C$</annotation>\n </semantics></math> such that <math>\n <semantics>\n <msup>\n <mi>C</mi>\n <mo>′</mo>\n </msup>\n <annotation>$C^\\prime$</annotation>\n </semantics></math> has the same lattice diameter as <math>\n <semantics>\n <mi>C</mi>\n <annotation>$C$</annotation>\n </semantics></math>. Similar structural results are obtained for lattice complete bodies. Moreover, various construction methods for lattice reduced and complete convex bodies are presented.","PeriodicalId":1,"journal":{"name":"Accounts of Chemical Research","volume":null,"pages":null},"PeriodicalIF":16.4000,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/jlms.12982","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Accounts of Chemical Research","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1112/jlms.12982","RegionNum":1,"RegionCategory":"化学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"CHEMISTRY, MULTIDISCIPLINARY","Score":null,"Total":0}

引用次数: 0

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.

Abstract Image

查看原文本刊更多论文

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。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

求助全文

约1分钟内获得全文求助全文

来源期刊

Accounts of Chemical Research 化学-化学综合

CiteScore

31.40

自引率

1.10%

发文量

312

审稿时长

2 months

期刊介绍： Accounts of Chemical Research presents short, concise and critical articles offering easy-to-read overviews of basic research and applications in all areas of chemistry and biochemistry. These short reviews focus on research from the author’s own laboratory and are designed to teach the reader about a research project. In addition, Accounts of Chemical Research publishes commentaries that give an informed opinion on a current research problem. Special Issues online are devoted to a single topic of unusual activity and significance. Accounts of Chemical Research replaces the traditional article abstract with an article "Conspectus." These entries synopsize the research affording the reader a closer look at the content and significance of an article. Through this provision of a more detailed description of the article contents, the Conspectus enhances the article's discoverability by search engines and the exposure for the research.