Journal of Convex Analysis 31 (2024), No. 3, 749--760
Copyright Heldermann Verlag 2024
On Poidge-Convexity
Xiangxiang Nie
School of Mathematical Sciences, Hebei Normal University, Shijiazhuang, P.R.China
xiangxiangnie@126.com
Liping Yuan
School of Mathematical Sciences, Hebei Normal University, Shijiazhuang, P.R.China
lpyuan@hebtu.edu.cn
Tudor Zamfirescu
(1) School of Mathematical Sciences, Hebei Normal University, Shijiazhuang, P.R.China
(2) Mathematical Institute, Roumanian Academy, Bucharest, Roumania
tuzamfirescu@gmail.com
[Abstract-pdf]
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.
Keywords: Poidge-convexity, unions of line-segments, complements, poidge-convex completion.
MSC: 52A01, 52A37.
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