About: Lower bounds for weak epsilon-nets and stair-convexity

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An Entity of Type : http://linked.opendata.cz/ontology/domain/vavai/Vysledek, within Data Space : linked.opendata.cz associated with source document(s)

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rdf:type	skos:Concept http://linked.opendata.cz/ontology/domain/vavai/Vysledek
rdfs:seeAlso	http://dx.doi.org/10.1007/s11856-011-0029-1
Description	A set N SUBSET OF ℝ d is called a weak ɛ-net (with respect to convex sets) for a finite X SUBSET OF ℝ d if N intersects every convex set C with \|X INTERSECTION C\| GREATER-THAN OR EQUAL TO ɛ\|X\|. For every fixed d GREATER-THAN OR EQUAL TO 2 and every r GREATER-THAN OR EQUAL TO 1 we construct sets X SUBSET OF ℝ d for which every weak 1/r -net has at least Ω(r log dMINUS SIGN 1 r) points; this is the first superlinear lower bound for weak ɛ-nets in a fixed dimension. The construction is a stretched grid, and convexity in this grid can be analyzed using stair-convexity, a new variant of the usual notion of convexity. We also consider weak ɛ-nets for the diagonal of our stretched grid in ℝ d , d GREATER-THAN OR EQUAL TO 3, which is an %22intrinsically 1-dimensional%22 point set. In this case we exhibit slightly superlinear lower bounds (involving the inverse Ackermann function), showing that the upper bounds by Alon, Kaplan, Nivasch, Sharir and Smorodinsky (2008) are not far from the truth in the worst case. Using the stretched grid we also improve the upper bound for the so-called second selection lemma in the plane by a logarithmic factor. A set N SUBSET OF ℝ d is called a weak ɛ-net (with respect to convex sets) for a finite X SUBSET OF ℝ d if N intersects every convex set C with \|X INTERSECTION C\| GREATER-THAN OR EQUAL TO ɛ\|X\|. For every fixed d GREATER-THAN OR EQUAL TO 2 and every r GREATER-THAN OR EQUAL TO 1 we construct sets X SUBSET OF ℝ d for which every weak 1/r -net has at least Ω(r log dMINUS SIGN 1 r) points; this is the first superlinear lower bound for weak ɛ-nets in a fixed dimension. The construction is a stretched grid, and convexity in this grid can be analyzed using stair-convexity, a new variant of the usual notion of convexity. We also consider weak ɛ-nets for the diagonal of our stretched grid in ℝ d , d GREATER-THAN OR EQUAL TO 3, which is an %22intrinsically 1-dimensional%22 point set. In this case we exhibit slightly superlinear lower bounds (involving the inverse Ackermann function), showing that the upper bounds by Alon, Kaplan, Nivasch, Sharir and Smorodinsky (2008) are not far from the truth in the worst case. Using the stretched grid we also improve the upper bound for the so-called second selection lemma in the plane by a logarithmic factor. (en)
Title	Lower bounds for weak epsilon-nets and stair-convexity Lower bounds for weak epsilon-nets and stair-convexity (en)
skos:prefLabel	Lower bounds for weak epsilon-nets and stair-convexity Lower bounds for weak epsilon-nets and stair-convexity (en)
skos:notation	RIV/00216208:11320/11:10100316!RIV12-MSM-11320___
http://linked.open...avai/predkladatel	Matematicko-fyzikální fakulta
http://linked.open...avai/riv/aktivita	P Z
http://linked.open...avai/riv/aktivity	P(1M0545), Z(MSM0021620838)
http://linked.open...iv/cisloPeriodika	1
http://linked.open...vai/riv/dodaniDat	2012
http://linked.open...aciTvurceVysledku	Matoušek, Jiří
http://linked.open.../riv/druhVysledku	J - Článek v odborném periodiku
http://linked.open...iv/duvernostUdaju	S - Úplné a pravdivé údaje nepodléhající ochraně podle zvláštních právních předpisů
http://linked.open...titaPredkladatele	Univerzita Karlova v Praze / Matematicko-fyzikální fakulta
http://linked.open...dnocenehoVysledku	210036
http://linked.open...ai/riv/idVysledku	RIV/00216208:11320/11:10100316
http://linked.open...riv/jazykVysledku	eng - angličtina
http://linked.open.../riv/klicovaSlova	computational geometry; lower bounds; stair convexity; weak epsilon-net (en)
http://linked.open.../riv/klicoveSlovo	lower bounds stair convexity weak epsilon-net computational geometry
http://linked.open...odStatuVydavatele	IL - Stát Izrael
http://linked.open...ontrolniKodProRIV	[492471141F85]
http://linked.open...i/riv/nazevZdroje	Israel Journal of Mathematics
http://linked.open...in/vavai/riv/obor	BA
http://linked.open...ichTvurcuVysledku	1 (xsd:int)
http://linked.open...cetTvurcuVysledku	3 (xsd:int)
http://linked.open...vavai/riv/projekt	Institute for Theoretical Computer Science
http://linked.open...UplatneniVysledku	2011
http://linked.open...v/svazekPeriodika	182
http://linked.open...iv/tvurceVysledku	Matoušek, Jiří Bukh, Boris Nivasch, Gabriel
http://linked.open...ain/vavai/riv/wos	000289109300009
http://linked.open...n/vavai/riv/zamer	Moderní metody, struktury a systémy informatiky
issn	0021-2172
number of pages	30 (xsd:int)
http://bibframe.org/vocab/doi	10.1007/s11856-011-0029-1
http://localhost/t...ganizacniJednotka	11320
is http://linked.open...avai/riv/vysledek of	Lower bounds for weak epsilon-nets and stair-convexity Lower bounds for weak epsilon-nets and stair-convexity

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