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

Fix any $\lambda\in\complexes$. We say that a set $S\subseteq\complexes$ is $\lambda$-convex if, whenever $a$ and $b$ are in $S$, the point $(1-\lambda)a+\lambda b$ is also in $S$. If $S$ is also (topologically) closed, then we say that $S$ is $\lambda$-clonvex. We investigate the properties of $\lambda$-convex and $\lambda$-clonvex sets and prove a number of facts about them. Letting $R_\lambda\subseteq\complexes$ be the least $\lambda$-clonvex superset of ${0,1}$, we show that if $R_\lambda$ is convex in the usual sense, then $R_\lambda$ must be either $[0,1]$ or $\reals$ or $\complexes$, depending on $\lambda$. We investigate which $\lambda$ make $R_\lambda$ convex, derive a number of conditions equivalent to $R_\lambda$ being convex, give several conditions sufficient for $R_\lambda$ to be convex or not convex (in particular, $R_\lambda$ is either convex or discrete), and investigate the properties of some particular discrete $R_\lambda$, as well as other $\lambda$-convex sets. Our work combines elementary concepts and techniques from algebra and plane geometry.