Pub Date : 2023-11-16DOI: 10.1007/s00454-023-00573-2
Otfried Cheong, Xavier Goaoc, Andreas F. Holmsen
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 (varepsilon )-nets for k-flats and convex sets in (mathbb {R}^d), 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 (mathbb {R}^3), and we extend a theorem of Karasev and Montejano regarding colorful intersections and k-transversals.
{"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":"130 1","pages":""},"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}
Pub Date : 2023-08-12DOI: 10.1007/s00454-023-00541-w
Jörg Rambau
{"title":"The Visible-Volume Function of a Set of Cameras is Continuous, Piecewise Rational, Locally Lipschitz, and Semi-Algebraic in All Dimensions","authors":"Jörg Rambau","doi":"10.1007/s00454-023-00541-w","DOIUrl":"https://doi.org/10.1007/s00454-023-00541-w","url":null,"abstract":"","PeriodicalId":50574,"journal":{"name":"Discrete & Computational Geometry","volume":"70 1","pages":"1038 - 1058"},"PeriodicalIF":0.8,"publicationDate":"2023-08-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43661006","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}
Pub Date : 2023-05-16DOI: 10.1007/s00454-023-00485-1
D. Cohen-Steiner, A. Lieutier, J. Vuillamy
{"title":"Delaunay and Regular Triangulations as Lexicographic Optimal Chains","authors":"D. Cohen-Steiner, A. Lieutier, J. Vuillamy","doi":"10.1007/s00454-023-00485-1","DOIUrl":"https://doi.org/10.1007/s00454-023-00485-1","url":null,"abstract":"","PeriodicalId":50574,"journal":{"name":"Discrete & Computational Geometry","volume":"70 1","pages":"1 - 50"},"PeriodicalIF":0.8,"publicationDate":"2023-05-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45486294","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}
Pub Date : 2023-01-01DOI: 10.1007/s00454-022-00446-0
Georg Loho, Raman Sanyal
Bárány's colorful generalization of Carathéodory's Theorem combines geometrical and combinatorial constraints. Kalai-Meshulam (2005) and Holmsen (2016) generalized Bárány's theorem by replacing color classes with matroid constraints. In this note, we obtain corresponding results in tropical convexity, generalizing the Tropical Colorful Carathéodory Theorem of Gaubert-Meunier (2010). Our proof is inspired by geometric arguments and is reminiscent of matroid intersection. Moreover, we show that the topological approach fails in this setting. We also discuss tropical colorful linear programming and show that it is NP-complete. We end with thoughts and questions on generalizations to polymatroids, anti-matroids as well as examples and matroid simplicial depth.
{"title":"Tropical Carathéodory with Matroids.","authors":"Georg Loho, Raman Sanyal","doi":"10.1007/s00454-022-00446-0","DOIUrl":"https://doi.org/10.1007/s00454-022-00446-0","url":null,"abstract":"<p><p>Bárány's colorful generalization of Carathéodory's Theorem combines geometrical and combinatorial constraints. Kalai-Meshulam (2005) and Holmsen (2016) generalized Bárány's theorem by replacing color classes with matroid constraints. In this note, we obtain corresponding results in tropical convexity, generalizing the Tropical Colorful Carathéodory Theorem of Gaubert-Meunier (2010). Our proof is inspired by geometric arguments and is reminiscent of matroid intersection. Moreover, we show that the topological approach fails in this setting. We also discuss tropical colorful linear programming and show that it is NP-complete. We end with thoughts and questions on generalizations to polymatroids, anti-matroids as well as examples and matroid simplicial depth.</p>","PeriodicalId":50574,"journal":{"name":"Discrete & Computational Geometry","volume":"69 1","pages":"139-155"},"PeriodicalIF":0.8,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC9805987/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"10494381","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-01-01Epub Date: 2023-03-22DOI: 10.1007/s00454-022-00414-8
Katrin Casel, Henning Fernau, Alexander Grigoriev, Markus L Schmid, Sue Whitesides
Unit square visibility graphs (USV) are described by axis-parallel visibility between unit squares placed in the plane. If the squares are required to be placed on integer grid coordinates, then USV become unit square grid visibility graphs (USGV), an alternative characterisation of the well-known rectilinear graphs. We extend known combinatorial results for USGV and we show that, in the weak case (i.e., visibilities do not necessarily translate into edges of the represented combinatorial graph), the area minimisation variant of their recognition problem is -hard. We also provide combinatorial insights with respect to USV, and as our main result, we prove their recognition problem to be -hard, which settles an open question.
{"title":"Combinatorial Properties and Recognition of Unit Square Visibility Graphs.","authors":"Katrin Casel, Henning Fernau, Alexander Grigoriev, Markus L Schmid, Sue Whitesides","doi":"10.1007/s00454-022-00414-8","DOIUrl":"10.1007/s00454-022-00414-8","url":null,"abstract":"<p><p>Unit square visibility graphs (USV) are described by axis-parallel visibility between unit squares placed in the plane. If the squares are required to be placed on integer grid coordinates, then USV become unit square grid visibility graphs (USGV), an alternative characterisation of the well-known rectilinear graphs. We extend known combinatorial results for USGV and we show that, in the weak case (i.e., visibilities do not necessarily translate into edges of the represented combinatorial graph), the area minimisation variant of their recognition problem is <math><mrow><mspace></mspace><mrow><mi>N</mi><mi>P</mi></mrow><mspace></mspace></mrow></math>-hard. We also provide combinatorial insights with respect to USV, and as our main result, we prove their recognition problem to be <math><mrow><mspace></mspace><mrow><mi>N</mi><mi>P</mi></mrow><mspace></mspace></mrow></math>-hard, which settles an open question.</p>","PeriodicalId":50574,"journal":{"name":"Discrete & Computational Geometry","volume":"69 4","pages":"937-980"},"PeriodicalIF":0.8,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC10169907/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"10301082","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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