Some properties of sets in the plane closed under linear extrapolation by a fixed parameter

Some properties of sets in the plane closed under linear extrapolation by a fixed parameter Fenner, Stephen; Green, Frederic; Gurjar, R.; Homer, Steven 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. File last revised 29 Aug 2020 (this version, v6).