Bulletin of the
Korean Mathematical Society
BKMS

ISSN(Print) 1015-8634 ISSN(Online) 2234-3016

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Bull. Korean Math. Soc. 2017; 54(3): 799-816

Online first article November 11, 2016      Printed May 31, 2017

https://doi.org/10.4134/BKMS.b160317

Copyright © The Korean Mathematical Society.

On Bonnesen-style Aleksandrov-Fenchel inequalities in $\mathbb R^{n}$

Chunna Zeng

Vienna University of Technology

Abstract

In this paper, we investigate the Bonnesen-style Aleksandrov-Fenchel inequalities in $\mathbb R^{n},$ which are the generalization of known Bonne\-sen-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 \ref{thm3} 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

MSC numbers: 52A30, 52A39

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