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ON BONNESEN-STYLE ALEKSANDROV-FENCHEL INEQUALITIES IN ℝn

  • Received : 2016.04.13
  • Published : 2017.05.31

Abstract

In this paper, we investigate the Bonnesen-style Aleksandrov-Fenchel inequalities in ${\mathbb{R}}^n$, which are the generalization of known Bonnesen-style inequalities. We first define the i-th symmetric mixed homothetic deficit ${\Delta}_i(K,L)$ and its special case, the i-th Aleksandrov-Fenchel isoperimetric deficit ${\Delta}_i(K)$. Secondly, we obtain some lower bounds of (n - 1)-th Aleksandrov Fenchel isoperimetric deficit ${\Delta}_{n-1}(K)$. Theorem 4 strengthens Groemer's result. As direct consequences, the stronger isoperimetric inequalities are established when n = 2 and n = 3. Finally, the reverse Bonnesen-style Aleksandrov-Fenchel inequalities are obtained. As a consequence, the new reverse Bonnesen-style inequality is obtained.

Keywords

mixed volume;isoperimetric inequality;Bonnesen-style inequality;Aleksandrov-Fenchel inequality

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Acknowledgement

Supported by : Chinese Scholarship Council, Chongqing Educational Committee, CQNU