Abstract
Let $P$ be a finite set of points in the plane in general position, that is, no three points of $P$ are on a common line. We say that a set $H$ of five points from $P$ is a $5$-hole in~$P$ if $H$ is the vertex set of a convex $5$-gon containing no other points of~$P$. For a positive integer $n$, let $h_5(n)$ be the minimum number of 5-holes among all sets of $n$ points in the plane in general position. Despite many efforts in the last 30 years, the best known asymptotic lower and upper bounds for $h_5(n)$ have been of order $n)$ and~$O(n^2)$, respectively. We show that $h_5(n) = n4/5n)$, obtaining the first superlinear lower bound on $h_5(n)$. The following structural result, which might be of independent interest, is a crucial step in the proof of this lower bound. If a finite set $P$ of points in the plane in general position is partitioned by a line $ into two subsets, each of size at least 5 and not in convex position, then $ intersects the convex hull of some 5-hole in~$P$. The proof of this result is computer-assisted.
Originalsprache | englisch |
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Titel | 33rd International Symposium on Computational Geometry (SoCG 2017) |
Redakteure/-innen | Boris Aronov, Matthew J. Katz |
Herausgeber (Verlag) | Schloss Dagstuhl - Leibniz-Zentrum für Informatik |
Seiten | 8:1-8:16 |
Band | 77 |
ISBN (elektronisch) | 978-395977038-5 |
DOIs | |
Publikationsstatus | Veröffentlicht - 2017 |
Veranstaltung | 33rd International Symposium on Computational Geometry: SoCG 2017 - The University of Queensland, St Lucia, Brisbane, Australien Dauer: 4 Juli 2017 → 7 Juli 2017 http://socg2017.smp.uq.edu.au/ |
Publikationsreihe
Name | Leibniz International Proceedings in Informatics (LIPIcs) |
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Herausgeber (Verlag) | Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik |
Konferenz
Konferenz | 33rd International Symposium on Computational Geometry |
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Kurztitel | SoCG |
Land/Gebiet | Australien |
Ort | Brisbane |
Zeitraum | 4/07/17 → 7/07/17 |
Internetadresse |