{"title":"Higher rank antipodality","authors":"Márton Naszódi, Zsombor Szilágyi, 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 .
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.