ON BONNESEN-STYLE SYMMETRIC MIXED ISOHOMOTHETIC INEQUALITY IN R2

doi:10.1007/s10473-019-0510-1

数学物理学报(英文版) ›› 2019, Vol. 39 ›› Issue (5): 1319-1329.doi: 10.1007/s10473-019-0510-1

ON BONNESEN-STYLE SYMMETRIC MIXED ISOHOMOTHETIC INEQUALITY IN R²

王媛媛¹, 王星星², 曾春娜³

1. College of Science, Wuhan University of Science and Technology, Wuhan 430081, China;
2. School of Mathematics and Statistics, Shanghai Lixin University of Accounting and Finance, Shanghai 201620, China;
3. School of Mathematical Sciences, Chongqing Normal University, Chongqing 401331, China

收稿日期:2018-01-05 修回日期:2019-01-10 出版日期:2019-10-25 发布日期:2019-11-11
通讯作者: Chunna ZENG E-mail:zengchn@163.com
作者简介:Yuanyuan WANG,E-mail:wangyuanyuan@wust.edu.cn;Xingxing WANG,E-mail:m13098792429@163.com
基金资助:
The first author is supported in part by the National Natural Science Foundation of China (11801048), the Natural Science Foundation Project of CSTC (cstc2017jcyjAX0022) and Innovation Support Program for Chongqing overseas Returnees (cx2018034).

ON BONNESEN-STYLE SYMMETRIC MIXED ISOHOMOTHETIC INEQUALITY IN R²

Yuanyuan WANG¹, Xingxing WANG², Chunna ZENG³

1. College of Science, Wuhan University of Science and Technology, Wuhan 430081, China;
2. School of Mathematics and Statistics, Shanghai Lixin University of Accounting and Finance, Shanghai 201620, China;
3. School of Mathematical Sciences, Chongqing Normal University, Chongqing 401331, China

Received:2018-01-05 Revised:2019-01-10 Online:2019-10-25 Published:2019-11-11
Contact: Chunna ZENG E-mail:zengchn@163.com
Supported by:
The first author is supported in part by the National Natural Science Foundation of China (11801048), the Natural Science Foundation Project of CSTC (cstc2017jcyjAX0022) and Innovation Support Program for Chongqing overseas Returnees (cx2018034).

摘要/Abstract

摘要： In this paper, we investigate the translative containment measure for a convex domain K_i to contain, or to be contained in the homothetic copy of another convex domain tK_j(t ≥ 0). Via the formulas of translative Blaschke and Poincaré in integral formula, we obtain a Bonnesen-style symmetric mixed isohomothetic inequality. The Bonnesen-style symmetric mixed isohomothetic inequality obtained is known as Bonnesen-style inequality if one of the domains is a disc. As a direct consequence, we attain an inequality which strengthen the result proved by Bonnesen, Blaschké and Flanders. Furthermore, by the containment measure and Blaschke's rolling theorem, we obtain the reverse Bonnesen-style symmetric mixed isohomothetic inequalities. These inequalities are the analogues of the known Bottema's result in 1933.

关键词: translative containment measure, isoperimetric inequality, Bonnesen-style inequality, Bonnesen-style symmetric mixed isohomothetic inequality, reverse Bonessen-style symmetric mixed isohomothetic inequality

Abstract: In this paper, we investigate the translative containment measure for a convex domain K_i to contain, or to be contained in the homothetic copy of another convex domain tK_j(t ≥ 0). Via the formulas of translative Blaschke and Poincaré in integral formula, we obtain a Bonnesen-style symmetric mixed isohomothetic inequality. The Bonnesen-style symmetric mixed isohomothetic inequality obtained is known as Bonnesen-style inequality if one of the domains is a disc. As a direct consequence, we attain an inequality which strengthen the result proved by Bonnesen, Blaschké and Flanders. Furthermore, by the containment measure and Blaschke's rolling theorem, we obtain the reverse Bonnesen-style symmetric mixed isohomothetic inequalities. These inequalities are the analogues of the known Bottema's result in 1933.

Key words: translative containment measure, isoperimetric inequality, Bonnesen-style inequality, Bonnesen-style symmetric mixed isohomothetic inequality, reverse Bonessen-style symmetric mixed isohomothetic inequality

中图分类号:

52A20

王媛媛, 王星星, 曾春娜. ON BONNESEN-STYLE SYMMETRIC MIXED ISOHOMOTHETIC INEQUALITY IN R²[J]. 数学物理学报(英文版), 2019, 39(5): 1319-1329.

Yuanyuan WANG, Xingxing WANG, Chunna ZENG. ON BONNESEN-STYLE SYMMETRIC MIXED ISOHOMOTHETIC INEQUALITY IN R²[J]. Acta mathematica scientia,Series B, 2019, 39(5): 1319-1329.

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ON BONNESEN-STYLE SYMMETRIC MIXED ISOHOMOTHETIC INEQUALITY IN R²

ON BONNESEN-STYLE SYMMETRIC MIXED ISOHOMOTHETIC INEQUALITY IN R²

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