Holes in 2-convex point sets

Result code in IS VaVaI
<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F00216208%3A11320%2F18%3A10384911" target="_blank" >RIV/00216208:11320/18:10384911 - isvavai.cz</a>
Result on the web
<a href="https://doi.org/10.1016/j.comgeo.2018.06.002" target="_blank" >https://doi.org/10.1016/j.comgeo.2018.06.002</a>
DOI - Digital Object Identifier
<a href="http://dx.doi.org/10.1016/j.comgeo.2018.06.002" target="_blank" >10.1016/j.comgeo.2018.06.002</a>

Result language
angličtina
Original language name
Holes in 2-convex point sets
Original language description
Let S be a set of n points in the plane in general position (no three points from S are collinear). For a positive integer k, a k-hole in S is a convex polygon with k vertices from S and no points of S in its interior. For a positive integer l, a simple polygon P is l-convex if no straight line intersects the interior of P in more than l connected components. A point set S is l-convex if there exists an l-convex polygonization of S. Considering a typical Erdős-Szekeres-type problem, we show that every 2-convex point set of size n contains an Omega(log(n))-hole. In comparison, it is well known that there exist arbitrarily large point sets in general position with no 7-hole. Further, we show that our bound is tight by constructing 2-convex point sets in which every hole has size O(log(n)).
Czech name
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Czech description
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Type
J<sub>imp</sub> - Article in a specialist periodical, which is included in the Web of Science database
CEP classification
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OECD FORD branch
10101 - Pure mathematics

Project
Result was created during the realization of more than one project. More information in the Projects tab.
Continuities
P - Projekt vyzkumu a vyvoje financovany z verejnych zdroju (s odkazem do CEP)

Publication year
2018
Confidentiality
S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

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