On Bonnesen-style Aleksandrov-Fenchel inequalities in $\mathbb R^{n}$
Bull. Korean Math. Soc. 2017 Vol. 54, No. 3, 799-816
Published online November 11, 2016
Printed May 31, 2017
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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