ON THREE MEASURES OF NON-CONVEXITY

Result code in IS VaVaI
<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F00216224%3A14310%2F17%3A00107867" target="_blank" >RIV/00216224:14310/17:00107867 - isvavai.cz</a>
Result on the web
<a href="https://arxiv.org/pdf/1410.0407.pdf" target="_blank" >https://arxiv.org/pdf/1410.0407.pdf</a>
DOI - Digital Object Identifier
<a href="http://dx.doi.org/10.1007/s11856-017-1467-1" target="_blank" >10.1007/s11856-017-1467-1</a>

Result language
angličtina
Original language name
ON THREE MEASURES OF NON-CONVEXITY
Original language description
The invisibility graph I (X) of a set X subset of R-d is a (possibly infinite) graph whose vertices are the points of X and two vertices are connected by an edge if and only if the straight-line segment connecting the two corresponding points is not fully contained in X. We consider the following three parameters of a set X: the clique number omega(I(X)), the chromatic number chi(I(X)) and the convexity number gamma(X), which is the minimum number of convex subsets of X that cover X. We settle a conjecture of Matousek and Valtr claiming that for every planar set X, gamma(X) can be bounded in terms of chi(I( X)). As a part of the proof we show that a disc with n one-point holes near its boundary has chi(I(X)) >= log log(n) but omega (I(X)) = 3. We also find sets X in R-5 with chi(X) = 2, but gamma( X) arbitrarily large.
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
10100 - 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)<br>S - Specificky vyzkum na vysokych skolach

Publication year
2017
Confidentiality
S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

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