{"title":"Some New Results on Geometric Transversals","authors":"Otfried Cheong, Xavier Goaoc, Andreas F. Holmsen","doi":"10.1007/s00454-023-00573-2","DOIUrl":"https://doi.org/10.1007/s00454-023-00573-2","url":null,"abstract":"<p>We investigate a number of questions, problems, and conjectures related to geometric transversal theory. Among our results we disprove a conjecture of Bárány and Kalai regarding weak <span>(varepsilon )</span>-nets for <i>k</i>-flats and convex sets in <span>(mathbb {R}^d)</span>, and we prove a conjecture of Arocha, Bracho, and Montejano regarding a colorful version of the Goodman–Pollack–Wenger transversal theorem. We also investigate the connected components of the space of line transversals to pairwise disjoint convex sets in <span>(mathbb {R}^3)</span>, and we extend a theorem of Karasev and Montejano regarding colorful intersections and <i>k</i>-transversals.</p>","PeriodicalId":50574,"journal":{"name":"Discrete & Computational Geometry","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2023-11-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138507782","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Threshold Phenomena for Random Cones","authors":"D. Hug, R. Schneider","doi":"10.1007/s00454-021-00323-2","DOIUrl":"https://doi.org/10.1007/s00454-021-00323-2","url":null,"abstract":"","PeriodicalId":50574,"journal":{"name":"Discrete & Computational Geometry","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2021-08-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141210449","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Improvement on the Crossing Number of Crossing-Critical Graphs","authors":"J. Barát, G. Tóth","doi":"10.1007/s00454-020-00264-2","DOIUrl":"https://doi.org/10.1007/s00454-020-00264-2","url":null,"abstract":"","PeriodicalId":50574,"journal":{"name":"Discrete & Computational Geometry","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2020-12-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141225352","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"On the Number of Weakly Connected Subdigraphs in Random kNN Digraphs","authors":"Selim Bahadır, E. Ceyhan","doi":"10.1007/s00454-020-00218-8","DOIUrl":"https://doi.org/10.1007/s00454-020-00218-8","url":null,"abstract":"","PeriodicalId":50574,"journal":{"name":"Discrete & Computational Geometry","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2020-07-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80824581","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
H. Bergold, S. Felsner, M. Scheucher, Felix Schröder, R. Steiner
{"title":"Topological Drawings Meet Classical Theorems from Convex Geometry","authors":"H. Bergold, S. Felsner, M. Scheucher, Felix Schröder, R. Steiner","doi":"10.1007/s00454-022-00408-6","DOIUrl":"https://doi.org/10.1007/s00454-022-00408-6","url":null,"abstract":"","PeriodicalId":50574,"journal":{"name":"Discrete & Computational Geometry","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2020-05-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141203045","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}