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. 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: In this paper, we investigate the translative containment measure for a convex domain Ki to contain, or to be contained in the homothetic copy of another convex domain tKj(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
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