Acta mathematica scientia,Series A ›› 2021, Vol. 41 ›› Issue (4): 989-996.

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

Yan Zhang1(),Chunna Zeng1,*(),Xingxing Wang2()   

  1. 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;;
  • 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)


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

CLC Number: 

  • O186.5