On Poidge-Convexity

Let $\mathcal{F}$ be a family of sets in $\mathbb{R}^d\ (\mathrm{always}\ d\geq 2)$. A set $M\subset\mathbb{R}^d$ is called {\it $\mathcal{F}$-convex}, if for any pair of distinct points $x, y \in M$, there is a set $F\in \mathcal{F}$ such that $x, y \in F$ and $F \subset M$. We obtain the poidge-convexity, when $\mathcal{F}$ consists of all unions $\{x\}\cup \sigma$, called {\it poidges}, where $x$ is a point, $\sigma$ a line-segment, and $\mathrm{ conv}(\{x\}\cup \sigma)$ a right triangle. In this paper we first present several new results on the poidge-convexity of various sets, such as unions of line-segments, fans, cones and cylinders, complements of some given sets and not simply connected sets. Then, we investigate the poidge-convex completion of compact convex sets, trying to determine the minimal number of points necessary to be added to make them poidge-convex.