Some properties of sets in the plane closed under linear extrapolation by a fixed parameter
Files
First author draft
Date
2012-12-12
DOI
Authors
Fenner, Stephen
Green, Frederic
Gurjar, R.
Homer, Steven
Version
First author draft
OA Version
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.
Abstract
Fix any 𝛌 ⊆ ℂ. We say that a set S ⊆ ℂ 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 ⊆ ℂ 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 ℂ, 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.
Description
File last revised 29 Aug 2020 (this version, v6).