On poidge-convexity (Q6620683)

Let \(\mathscr F\) be a family of sets in \({\mathbb R}^d\) with \(d\geq 2\). Say that \(M\subset {\mathbb R}^d\) is \({\mathbb R}^d\)-convex provided that for every two \(x, y\in M\) there is \(F\in \mathscr F\) satisfying \(x, y\in F\) and \(F\subset M\). In case \(\mathscr F\) consists of all so-called poidges \(\{x\}\cup \sigma\), with \(\{x\}\) a singleton and \(\sigma\) a line segment, and \(\mathrm{conv}(\{x\}\cup \sigma)\) a right triangle. \N\NThe authors collect the elementary properties of the poidge-convexity of some collections of sets and address the possibility of the poidge-convex completion of compact convex sets by ading a few points.