Acta mathematica scientia,Series B ›› 2025, Vol. 45 ›› Issue (1): 3-15.doi: 10.1007/s10473-025-0101-2

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Γ-CONVEXITY

Zhouqin Jia1, Wenzhi Liu1 Liping Yuan2,3,*, Tudor Zamfirescu4,5,6   

  1. 1. School of Mathematical Sciences, Hebei Normal University, Shijiazhuang 050024, China;
    2. School of Mathematical Sciences, Hebei Normal University, Shijiazhuang 050024, China;
    3. Hebei Key Laboratory of Computational Mathematics and Applications, Shijiazhuang 050024, China;
    4. School of Mathematical Sciences, Hebei Normal University, Shijiazhuang 050024, China;
    5. Fachbereich Mathematik, Technische Universität Dortmund, Dortmund 44221, Germany;
    6. Roumanian Academy, Bucharest 014700, Roumania
  • Received:2024-06-11 Revised:2024-11-03 Published:2025-02-06
  • Contact: *Liping Yuan, E-mail,: lpyuan@hebtu.edu.cn
  • About author:Zhouqin Jia, E-mail,: 15031320996@163.com; Wenzhi Liu, E-mail,: wenzhiliu0601@163.com; Tudor Zamfirescu, E-mail,: tuzamfirescu@gmail.com
  • Supported by:
    NSFC (12271139, 11871192); the High-end Foreign Experts Recruitment Program of People's Republic of China (G2023003003L); the Program for Foreign experts of Hebei Province; the Natural Science Foundation of Hebei Province (A2023205045); the Special Project on Science and Technology Research and Development Platforms, Hebei Province (22567610H) and the program for 100 Foreign Experts Plan of Hebei Province.

Abstract: Let F be a family of sets in Rd (always d2). A set MRd is called F-convex, if for any pair of distinct points x,yM, there is a set FF such that x,yF and FM. We obtain the Γ-convexity, when F consists of Γ-paths. A Γ-path is the union of both shorter sides of an isosceles right triangle. In this paper we first characterize some Γ-convex sets, bounded or unbounded, including triangles, regular polygons, subsets of balls, right cylinders and cones, unbounded planar closed convex sets, etc. Then, we investigate the Γ-starshaped sets, and provide some conditions for a fan, a spherical sector and a right cylinder to be Γ-starshaped. Finally, we study the Γ-triple-convexity, which is a discrete generalization of Γ-convexity, and provide characterizations for all the 4-point sets, some 5-point sets and Zd to be Γ-triple-convex.

Key words: Γ-convexity, Γ-starshaped sets, Γ-triple-convexity

CLC Number: 

  • 52A01
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