ON BONNESEN-STYLE SYMMETRIC MIXED ISOHOMOTHETIC INEQUALITY IN R2

doi:10.1007/s10473-019-0510-1

Acta mathematica scientia,Series B ›› 2019, Vol. 39 ›› Issue (5): 1319-1329.doi: 10.1007/s10473-019-0510-1

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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).

Abstract

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

CLC Number:

52A20

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

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