dc.contributor.author Fenner, Stephen en_US dc.contributor.author Green, Frederic en_US dc.contributor.author Gurjar, R. en_US dc.contributor.author Homer, Steven en_US dc.date.accessioned 2021-11-03T18:21:21Z dc.date.available 2021-11-03T18:21:21Z dc.date.issued 2012-12-12 dc.identifier http://arxiv.org/abs/1212.2889v1 dc.identifier.citation S. 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.uri https://hdl.handle.net/2144/43263 dc.description File last revised 29 Aug 2020 (this version, v6). en_US dc.description.abstract Fix 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 βor 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.iso en_US dc.title Some properties of sets in the plane closed under linear extrapolation by a fixed parameter en_US dc.type Article en_US dc.description.version First author draft en_US pubs.elements-source manual-entry en_US pubs.notes 53 pages, 19 figures en_US pubs.organisational-group Boston University en_US pubs.organisational-group Boston University, College of Arts & Sciences en_US pubs.organisational-group Boston University, College of Arts & Sciences, Department of Computer Science en_US pubs.publication-status Unpublished en_US dc.identifier.mycv 186485
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