Lower bounds for the number of small convex k-holes

Let S be a set of n points in the plane in general position, that is, no three points of S are on a line. We consider an Erdos-type question on the least number h(k)(n) of convex k-holes in S, and give improved lower bounds on h(k)(n), for 3 <= k <= 5. Specifically, we show that h(3)(n) >=...

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Autores: Aichholzer, Oswin, Fabila Monroy, Ruy, Hackl, Thomas, Huemer, Clemens|||0000-0001-7557-0823, Pilz, Alexander, Vogtenhuber, Birgit
Tipo de recurso: artículo
Fecha de publicación:2014
País:España
Institución:Universitat Politècnica de Catalunya (UPC)
Repositorio:UPCommons. Portal del coneixement obert de la UPC
Idioma:inglés
OAI Identifier:oai:upcommons.upc.edu:2117/26660
Acceso en línea:https://hdl.handle.net/2117/26660
https://dx.doi.org/10.1016/j.comgeo.2013.12.002
Access Level:acceso abierto
Palabra clave:Discrete geometry
Combinatorial geometry
Empty convex polygon
Erdos-type problem
Counting
PLANAR POINT SETS
EMPTY
POLYGONS
THEOREM
Geometria combinatòria
Àrees temàtiques de la UPC::Matemàtiques i estadística::Geometria::Geometria convexa i discreta
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spelling Lower bounds for the number of small convex k-holesAichholzer, OswinFabila Monroy, RuyHackl, ThomasHuemer, Clemens|||0000-0001-7557-0823Pilz, AlexanderVogtenhuber, BirgitDiscrete geometryCombinatorial geometryEmpty convex polygonErdos-type problemCountingPLANAR POINT SETSEMPTYPOLYGONSTHEOREMGeometria combinatòriaÀrees temàtiques de la UPC::Matemàtiques i estadística::Geometria::Geometria convexa i discretaLet S be a set of n points in the plane in general position, that is, no three points of S are on a line. We consider an Erdos-type question on the least number h(k)(n) of convex k-holes in S, and give improved lower bounds on h(k)(n), for 3 <= k <= 5. Specifically, we show that h(3)(n) >= n(2) - 32n/7 + 22/7, h(4)(n) >= n(2)/2 - 9n/4 - o(n), and h(5)(n) >= 3n/4 - o(n). We further settle several questions on sets of 12 points posed by Dehnhardt in 1987. (C) 2013 Elsevier B.V. All rights reserved.20142014-07-0120152015-03-11journal articlehttp://purl.org/coar/resource_type/c_6501AMhttp://purl.org/coar/version/c_ab4af688f83e57aainfo:eu-repo/semantics/articleapplication/pdfhttps://hdl.handle.net/2117/26660https://dx.doi.org/10.1016/j.comgeo.2013.12.002reponame:UPCommons. Portal del coneixement obert de la UPCinstname:Universitat Politècnica de Catalunya (UPC)Inglésengopen accesshttp://purl.org/coar/access_right/c_abf2Attribution-NonCommercial-NoDerivs 3.0 Spainhttp://creativecommons.org/licenses/by-nc-nd/3.0/es/info:eu-repo/semantics/openAccessoai:upcommons.upc.edu:2117/266602026-05-27T15:37:01Z
dc.title.none.fl_str_mv Lower bounds for the number of small convex k-holes
title Lower bounds for the number of small convex k-holes
spellingShingle Lower bounds for the number of small convex k-holes
Aichholzer, Oswin
Discrete geometry
Combinatorial geometry
Empty convex polygon
Erdos-type problem
Counting
PLANAR POINT SETS
EMPTY
POLYGONS
THEOREM
Geometria combinatòria
Àrees temàtiques de la UPC::Matemàtiques i estadística::Geometria::Geometria convexa i discreta
title_short Lower bounds for the number of small convex k-holes
title_full Lower bounds for the number of small convex k-holes
title_fullStr Lower bounds for the number of small convex k-holes
title_full_unstemmed Lower bounds for the number of small convex k-holes
title_sort Lower bounds for the number of small convex k-holes
dc.creator.none.fl_str_mv Aichholzer, Oswin
Fabila Monroy, Ruy
Hackl, Thomas
Huemer, Clemens|||0000-0001-7557-0823
Pilz, Alexander
Vogtenhuber, Birgit
author Aichholzer, Oswin
author_facet Aichholzer, Oswin
Fabila Monroy, Ruy
Hackl, Thomas
Huemer, Clemens|||0000-0001-7557-0823
Pilz, Alexander
Vogtenhuber, Birgit
author_role author
author2 Fabila Monroy, Ruy
Hackl, Thomas
Huemer, Clemens|||0000-0001-7557-0823
Pilz, Alexander
Vogtenhuber, Birgit
author2_role author
author
author
author
author
dc.subject.none.fl_str_mv Discrete geometry
Combinatorial geometry
Empty convex polygon
Erdos-type problem
Counting
PLANAR POINT SETS
EMPTY
POLYGONS
THEOREM
Geometria combinatòria
Àrees temàtiques de la UPC::Matemàtiques i estadística::Geometria::Geometria convexa i discreta
topic Discrete geometry
Combinatorial geometry
Empty convex polygon
Erdos-type problem
Counting
PLANAR POINT SETS
EMPTY
POLYGONS
THEOREM
Geometria combinatòria
Àrees temàtiques de la UPC::Matemàtiques i estadística::Geometria::Geometria convexa i discreta
description Let S be a set of n points in the plane in general position, that is, no three points of S are on a line. We consider an Erdos-type question on the least number h(k)(n) of convex k-holes in S, and give improved lower bounds on h(k)(n), for 3 <= k <= 5. Specifically, we show that h(3)(n) >= n(2) - 32n/7 + 22/7, h(4)(n) >= n(2)/2 - 9n/4 - o(n), and h(5)(n) >= 3n/4 - o(n). We further settle several questions on sets of 12 points posed by Dehnhardt in 1987. (C) 2013 Elsevier B.V. All rights reserved.
publishDate 2014
dc.date.none.fl_str_mv 2014
2014-07-01
2015
2015-03-11
dc.type.none.fl_str_mv journal article
http://purl.org/coar/resource_type/c_6501
AM
http://purl.org/coar/version/c_ab4af688f83e57aa
dc.type.openaire.fl_str_mv info:eu-repo/semantics/article
format article
dc.identifier.none.fl_str_mv https://hdl.handle.net/2117/26660
https://dx.doi.org/10.1016/j.comgeo.2013.12.002
url https://hdl.handle.net/2117/26660
https://dx.doi.org/10.1016/j.comgeo.2013.12.002
dc.language.none.fl_str_mv Inglés
eng
language_invalid_str_mv Inglés
language eng
dc.rights.none.fl_str_mv open access
http://purl.org/coar/access_right/c_abf2
Attribution-NonCommercial-NoDerivs 3.0 Spain
http://creativecommons.org/licenses/by-nc-nd/3.0/es/
dc.rights.openaire.fl_str_mv info:eu-repo/semantics/openAccess
rights_invalid_str_mv open access
http://purl.org/coar/access_right/c_abf2
Attribution-NonCommercial-NoDerivs 3.0 Spain
http://creativecommons.org/licenses/by-nc-nd/3.0/es/
eu_rights_str_mv openAccess
dc.format.none.fl_str_mv application/pdf
dc.source.none.fl_str_mv reponame:UPCommons. Portal del coneixement obert de la UPC
instname:Universitat Politècnica de Catalunya (UPC)
instname_str Universitat Politècnica de Catalunya (UPC)
reponame_str UPCommons. Portal del coneixement obert de la UPC
collection UPCommons. Portal del coneixement obert de la UPC
repository.name.fl_str_mv
repository.mail.fl_str_mv
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