Higher rank antipodality

IF 0.8 3区 数学 Q2 MATHEMATICS
Mathematika Pub Date : 2025-09-08 DOI:10.1112/mtk.70046
Márton Naszódi, Zsombor Szilágyi, Mihály Weiner
{"title":"Higher rank antipodality","authors":"Márton Naszódi,&nbsp;Zsombor Szilágyi,&nbsp;Mihály Weiner","doi":"10.1112/mtk.70046","DOIUrl":null,"url":null,"abstract":"<p>Motivated by general probability theory, we say that the set <span></span><math></math> in <span></span><math></math> is <i>antipodal of rank</i> <span></span><math></math>, if for any <span></span><math></math> elements <span></span><math></math>, there is an affine map from <span></span><math></math> to the <span></span><math></math>-dimensional simplex <span></span><math></math> that maps <span></span><math></math> bijectively onto the <span></span><math></math> vertices of <span></span><math></math>. For <span></span><math></math>, it coincides with the well-studied notion of (pairwise) antipodality introduced by Klee. We consider the following natural generalization of Klee's problem on antipodal sets: What is the maximum size of an antipodal set of rank <span></span><math></math> in <span></span><math></math>? We present a geometric characterization of antipodal sets of rank <span></span><math></math> and adapting the argument of Danzer and Grünbaum originally developed for the <span></span><math></math> case, we prove an upper bound which is exponential in the dimension. We show that this problem can be connected to a classical question in computer science on finding perfect hashes, and it provides a lower bound on the maximum size, which is also exponential in the dimension. By connecting rank-<span></span><math></math> antipodality to <span></span><math></math>-neighborly polytopes, we obtain another upper bound when <span></span><math></math>.</p>","PeriodicalId":18463,"journal":{"name":"Mathematika","volume":"71 4","pages":""},"PeriodicalIF":0.8000,"publicationDate":"2025-09-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://londmathsoc.onlinelibrary.wiley.com/doi/epdf/10.1112/mtk.70046","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Mathematika","FirstCategoryId":"100","ListUrlMain":"https://londmathsoc.onlinelibrary.wiley.com/doi/10.1112/mtk.70046","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0

Abstract

Motivated by general probability theory, we say that the set in is antipodal of rank , if for any elements , there is an affine map from to the -dimensional simplex that maps bijectively onto the vertices of . For , it coincides with the well-studied notion of (pairwise) antipodality introduced by Klee. We consider the following natural generalization of Klee's problem on antipodal sets: What is the maximum size of an antipodal set of rank in ? We present a geometric characterization of antipodal sets of rank and adapting the argument of Danzer and Grünbaum originally developed for the case, we prove an upper bound which is exponential in the dimension. We show that this problem can be connected to a classical question in computer science on finding perfect hashes, and it provides a lower bound on the maximum size, which is also exponential in the dimension. By connecting rank- antipodality to -neighborly polytopes, we obtain another upper bound when .

Abstract Image

Abstract Image

高阶反对性
根据一般概率论,我们说,如果对于任何元素,存在一个仿射映射,映射到的-维单纯形的顶点上,则集合是秩对映的。因为,它与Klee提出的(成对)反极性的概念相吻合。我们考虑对映集上的Klee问题的以下自然推广:秩为的对映集的最大大小是多少?我们给出了秩对映集的几何表征,并采用了Danzer和grnbaum最初针对这种情况提出的论点,证明了一个上界在维数上是指数的。我们证明了这个问题可以与计算机科学中寻找完美哈希的经典问题联系起来,并且它提供了最大大小的下界,它在维度上也是指数级的。通过将秩对对与邻多边形连接起来,得到了另一个上界。
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
Mathematika
Mathematika MATHEMATICS, APPLIED-MATHEMATICS
CiteScore
1.40
自引率
0.00%
发文量
60
审稿时长
>12 weeks
期刊介绍: Mathematika publishes both pure and applied mathematical articles and has done so continuously since its founding by Harold Davenport in the 1950s. The traditional emphasis has been towards the purer side of mathematics but applied mathematics and articles addressing both aspects are equally welcome. The journal is published by the London Mathematical Society, on behalf of its owner University College London, and will continue to publish research papers of the highest mathematical quality.
×
引用
GB/T 7714-2015
复制
MLA
复制
APA
复制
导出至
BibTeX EndNote RefMan NoteFirst NoteExpress
×
提示
您的信息不完整,为了账户安全,请先补充。
现在去补充
×
提示
您因"违规操作"
具体请查看互助需知
我知道了
×
提示
确定
请完成安全验证×
copy
已复制链接
快去分享给好友吧!
我知道了
右上角分享
点击右上角分享
0
联系我们:info@booksci.cn Book学术提供免费学术资源搜索服务,方便国内外学者检索中英文文献。致力于提供最便捷和优质的服务体验。 Copyright © 2023 布克学术 All rights reserved.
京ICP备2023020795号-1
ghs 京公网安备 11010802042870号
Book学术文献互助
Book学术文献互助群
群 号:604180095
Book学术官方微信