$\mathbb{R}^3$中四面体的几个新Bonnesen型不等式

数学物理学报 ›› 2021, Vol. 41 ›› Issue (4): 989-996.

$\mathbb{R}^3$中四面体的几个新Bonnesen型不等式

张燕¹(),曾春娜^1,*(),王星星²()

¹ 重庆师范大学数学科学学院重庆 401331
² 上海立信会计金融学院统计与数学学院上海 201620

收稿日期:2020-09-09 出版日期:2021-08-26 发布日期:2021-08-09
通讯作者: 曾春娜 E-mail:2279282928@qq.com;zengchn@163.com;m13098792429@163.com
作者简介:张燕, E-mail: 2279282928@qq.com|王星星, E-mail: m13098792429@163.com
基金资助:
国家自然科学基金(11801048);重庆市自然科学基金(cstc2020jcyj-msxmX0609);重庆市留学人员创新创业支持计划(cx2018034);重庆市留学人员创新创业支持计划(cx2019155);重庆市教育委员会科学技术研究项目(KJQN201900530)

Some New Bonnesen-Type Inequalities of the Tetrahedron in $\mathbb{R}^3$

Yan Zhang¹(),Chunna Zeng^1,*(),Xingxing Wang²()

¹ School of Mathematical Sciences, Chongqing Normal University, Chongqing 401331
² School of Mathematics and Statistics, Shanghai Lixin University of Accounting and Finance, Shanghai 201620

Received:2020-09-09 Online:2021-08-26 Published:2021-08-09
Contact: Chunna Zeng E-mail:2279282928@qq.com;zengchn@163.com;m13098792429@163.com
Supported by:
the NSFC(11801048);the NSF of Chongqin(cstc2020jcyj-msxmX0609);the Venture Innovation Support Program for Chongqing Overseas Returnees(cx2018034);the Venture Innovation Support Program for Chongqing Overseas Returnees(cx2019155);the Technology Research Foundation of Chongqing Educational Committee(KJQN201900530)

摘要/Abstract

摘要：

离散的等周问题在积分几何与凸几何中扮演着重要角色.等周亏格的稳定性可以由Bonnesen型不等式和逆Bonnesen型不等式来刻画.该文主要研究$\mathbb{R}^3$中四面体的Bonnesen型不等式和逆Bonnesen型不等式，获得了四面体的几个新的Bonnesen型不等式，并提供了不同于Sturm^[15]关于四面体的等周不等式的一种简化证明；最后获得了几个用四面体内切球半径以及外接球半径表示的逆Bonnesen型不等式.

关键词: 四面体, 等周亏格, Bonnesen型不等式, 逆Bonnesen型不等式

Abstract:

Discrete isoperimetric problems play an important role in integral geometry and convex geometry. The stability of isoperimetric deficit can be characterized by Bonnesen-type inequality and inverse Bonnesen-type inequality. In this paper, we study the Bonnesen-type inequality and the inverse Bonnesen-type inequality for Tetrahedra in $\mathbb{R}^3$. And we obtain several new Bonnesen-type inequalities for Tetrahedra. It provides a simplified proof which is different from the isoperimetric inequality for Tetrahedra in Sturm ^[15]; finally, four inverse Bonnesen-type inequalities in terms of the radius of the circumscribed sphere and the radius of the circumscribed sphere are obtained.

Key words: Tetrahedron, Isoperimetric deficit, Bonnesen-type inequality, Inverse Bonnesen-type inequality

中图分类号:

O186.5

张燕,曾春娜,王星星. $\mathbb{R}^3$中四面体的几个新Bonnesen型不等式[J]. 数学物理学报, 2021, 41(4): 989-996.

Yan Zhang,Chunna Zeng,Xingxing Wang. Some New Bonnesen-Type Inequalities of the Tetrahedron in $\mathbb{R}^3$[J]. Acta mathematica scientia,Series A, 2021, 41(4): 989-996.

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[1]	曾春娜,彭璐,马磊,王星星. $\mathbb{R}^3$中四面体的Bonnesen型等周不等式[J]. 数学物理学报, 2021, 41(2): 296-302.
[2]	张洪, 徐文学, 李泽清. 关于平面常宽凸集等周亏格的注记[J]. 数学物理学报, 2013, 33(6): 1162-1168.