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dc.contributor.authorFenner, Stephenen_US
dc.contributor.authorGreen, Fredericen_US
dc.contributor.authorGurjar, R.en_US
dc.contributor.authorHomer, Stevenen_US
dc.date.accessioned2021-11-03T18:21:21Z
dc.date.available2021-11-03T18:21:21Z
dc.date.issued2012-12-12
dc.identifierhttp://arxiv.org/abs/1212.2889v1
dc.identifier.citationS. Fenner, F. Green, R. Gurjar, S. Homer. 2012. "Some properties of sets in the plane closed under linear extrapolation by a fixed parameter." https://arxiv.org/abs/1212.2889v1.
dc.identifier.urihttps://hdl.handle.net/2144/43263
dc.descriptionFile last revised 29 Aug 2020 (this version, v6).en_US
dc.description.abstractFix any 𝛌 ⊆ β„‚. We say that a set S ⊆ β„‚ is 𝛌-convex if, whenever a and b are in S, the point (1- 𝛌)a + 𝛌b is also in S. If S is also (topologically) closed, then we say that S is 𝛌-clonvex. We investigate the properties of 𝛌-convex and 𝛌-clonvex sets and prove a number of facts about them. Letting R_𝛌 ⊆ β„‚ be the least 𝛌-clonvex superset of {0,1}, we show that if R_𝛌 is convex in the usual sense, then R_𝛌 must be either [0,1] or ℝor β„‚, depending on 𝛌. We investigate which 𝛌 make R_𝛌 convex, derive a number of conditions equivalent to R_𝛌 being convex, give several conditions sufficient for R_𝛌 to be convex or not convex in particular, R_𝛌 is either convex or discrete, and investigate the properties of some particular discrete R_𝛌, as well as other 𝛌-convex sets. Our work combines elementary concepts and techniques from algebra and plane geometry.en_US
dc.language.isoen_US
dc.titleSome properties of sets in the plane closed under linear extrapolation by a fixed parameteren_US
dc.typeArticleen_US
dc.description.versionFirst author draften_US
pubs.elements-sourcemanual-entryen_US
pubs.notes53 pages, 19 figuresen_US
pubs.organisational-groupBoston Universityen_US
pubs.organisational-groupBoston University, College of Arts & Sciencesen_US
pubs.organisational-groupBoston University, College of Arts & Sciences, Department of Computer Scienceen_US
pubs.publication-statusUnpublisheden_US
dc.identifier.mycv186485
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