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›› 2004, Vol. 21 ›› Issue (2): 153-163.DOI: 10.7523/j.issn.2095-6134.2004.2.002

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Several Inequalities in Differential Geometry and Their Generalizations

MA Hong-Bin1, SUN Zhen-Zu2   

  1. 1. Institute of Systems Science, AMSS, Chinese Academy of Sciences, Beijing 100080, China;
    2. Department of Mathematics and Systems Science, Zhengzhou University, Zhengzhou 450052, China
  • Received:2003-03-04 Revised:2003-06-02 Online:2004-03-19

Abstract:

Several inequalities are discussed. (1) For Schur. s inequality on convex curves of plane, we give a newanalytic proof for it, which maybe is simpler or clearer than known ones; we make further discussions by means ofintegral geometry and get more results. Moreover several related inequalities are put forward and proved. We alsopropose a conjecture which is generalization of Schur. s inequality in case of spherical surface. (2) For Fáry. s inequalityon closed curves of Euclidean space E3, we generalize it into spherical surface (i. e. surface with positiveconstant Gauss curvature) using method of moving frame. (3) For Fáry. s inequality on closed surface of Euclideanspace E3 : , we enhance it to using method of moving frame. Moreover thisinequality has been also generalized into 4-dimension case: Furthermore, a conjectureon further generalization to higher dimension case or general Euclidean space is proposed which requires furtherstudy.

Key words: Schur’s inequality, F?ry’s inequality, integral geometry, moving frame, Gauss curvature

CLC Number: