Abstract

Let F\mathcal{F} be a family of sets in Rd (always d≥2)\mathbb{R}^d\ (\mathrm{always}\ d\geq 2). A set M⊂RdM\subset\mathbb{R}^d is called F\mathcal{F}-convex, if for any pair of distinct points x,y∈Mx, y \in M, there is a set F∈FF\in \mathcal{F} such that x,y∈Fx, y \in F and F⊂MF \subset M. We obtain the poidge-convexity, when F\mathcal{F} consists of all unions {x}∪σ\{x\}\cup \sigma, called poidges, where xx is a point, σ\sigma a line-segment, and conv({x}∪σ)\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.

Author information

Contact details are reproduced from the original publication and may be historical.

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

Keywords

  • Poidge-convexity
  • unions of line-segments
  • complements
  • poidge-convex completion

Mathematics Subject Classification

52A01, 52A37

References

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Suggested citation

X. Nie, L. Yuan, T. Zamfirescu. “On Poidge-Convexity.” Journal of Convex Analysis 31 (2024), No. 3, 749–760. https://doi.org/10.68381/jca31036

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Published by Heldermann Verlag, 2024. Rights now held by Banach Press LLC.