2026-07-26T04:50:00Z https://drops.dagstuhl.de/oai

oai:drops-oai.dagstuhl.de:10426 2024-03-06T10:45:57Z ddc:004 open_access

Convex Polygons in Cartesian Products De Carufel, Jean-Lou Dumitrescu, Adrian Meulemans, Wouter Ophelders, Tim Pennarun, Claire Tóth, Csaba D. Verdonschot, Sander Erdős-Szekeres theorem Cartesian product convexity polyhedron recursive construction approximation algorithm We study several problems concerning convex polygons whose vertices lie in a Cartesian product of two sets of n real numbers (for short, grid). First, we prove that every such grid contains a convex polygon with Omega(log n) vertices and that this bound is tight up to a constant factor. We generalize this result to d dimensions (for a fixed d in N), and obtain a tight lower bound of Omega(log^{d-1}n) for the maximum number of points in convex position in a d-dimensional grid. Second, we present polynomial-time algorithms for computing the longest convex polygonal chain in a grid that contains no two points with the same x- or y-coordinate. We show that the maximum size of such a convex polygon can be efficiently approximated up to a factor of 2. Finally, we present exponential bounds on the maximum number of convex polygons in these grids, and for some restricted variants. These bounds are tight up to polynomial factors. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Jean-Lou De Carufel and Adrian Dumitrescu and Wouter Meulemans and Tim Ophelders and Claire Pennarun and Csaba D. Tóth and Sander Verdonschot 2019 Is Part Of LIPIcs, Volume 129, 35th International Symposium on Computational Geometry (SoCG 2019) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.SoCG.2019.22 urn:nbn:de:0030-drops-104267 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2019.22 eng https://creativecommons.org/licenses/by/3.0/legalcode