基于几何凸集的几何凸函数
内蒙古民族大学, 数理学院, 内蒙古通辽 028000
| 摘 要： | 几何凸函数是与凸函数平行的一个概念，通过一种变换或者不等式给出，它推广了凸函数的很多控制不等式理论和方法。随着几何凸集概念的提出，基于几何凸集的几何凸函数的性质被研究，但是关于几何凸集和几何凸函数的性质并未完善。本文通过文献研究法、分析综合法和归纳演绎法进一步探讨了几何凸集和几何凸函数的性质。首先在几何凸集概念的基础上，定义了集合的几何凸包和几何凸组合，给出了一个判定几何凸集的等价条件和一个几何凸组合的例子。然后利用上图给出了几何凸函数的等价定义，得到了几何凸函数的一个判定定理。同时，定义函数的几何凸包，利用函数的几何凸包得到了几何凸函数的另一个判定定理。最后，研究了几何凸函数在几种运算下保持几何凸性的不变性质以及几何凸函数的局部全局性质。本文的研究结果将进一步丰富几何凸性理论。 |
| 关 键 词： | 几何凸函数; 几何凸包; 几何凸组合; 上图 |
| DOI： | 10.57237/j.wjms.2022.01.003 |
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收稿日期: 2022-10-20
接受日期: 2022-11-15
在线出版日期: 2022-12-01
Geometric Convex Function Based on Geometric Convex Set
College of Mathematics and Physics, Inner Mongolia Minzu University, Tongliao 028000, China
| Abstract: | Geometric convex function is a parallel concept with convex function and is given by a transformation or an inequality, which generalizes the theory and method of control inequality for convex function. With the concept of geometric convex set being put forward, the properties of geometric convex function based on geometric convex set have been studied. But the properties of geometric convex set and geometric convex function are not perfect. In this paper, we further discuss the properties of geometric convex set and geometric convex function by literature research, analysis, synthesis, induction and deduction. Firstly, based on the concept of geometric convex set, the geometric convex hull and geometric convex combination of a set are defined, and an equivalent condition for judging geometric convex set and an example of geometric convex combination are given. Then, the equivalent definition of geometric convex function is given by using the epigraph, and a judgment theorem of geometric convex function is obtained. In addition, the geometric convex hull of the function is defined, and another judgment theorem of geometric convex function is also obtained by using geometric convex hull of the function. Finally, the invariance of geometric convexity under several operations and the local and global properties of geometric convex functions are studied. The results of this paper will further enrich the theory of geometric convexity. |
| Keywords: | Geometric Convex Function; Geometric Convex Hull; Geometric Convex Combination; Epigraph |
基金
| 1. | 内蒙古自治区直属高校基本科研业务费项目 (2022) |
| 2. | 内蒙古民族大学博士科研启动基金项目 (BS402). |
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