• Title, Summary, Keyword: the Bonnesen-style inequality

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ON BONNESEN-STYLE ALEKSANDROV-FENCHEL INEQUALITIES IN ℝn

  • Zeng, Chunna
    • Bulletin of the Korean Mathematical Society
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    • v.54 no.3
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    • pp.799-816
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    • 2017
  • 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.

SOME NEW BONNESEN-STYLE INEQUALITIES

  • Zhou, Jiazu;Xia, Yunwei;Zeng, Chunna
    • Journal of the Korean Mathematical Society
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    • v.48 no.2
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    • pp.421-430
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    • 2011
  • By evaluating the containment measure of one domain to contain another, we will derive some new Bonnesen-type inequalities (Theorem 2) via the method of integral geometry. We obtain Ren's sufficient condition for one domain to contain another domain (Theorem 4). We also obtain some new geometric inequalities. Finally we give a simplified proof of the Bottema's result.

ON THE ISOPERIMETRIC DEFICIT UPPER LIMIT

  • Zhou, Jiazu;Ma, Lei;Xu, Wenxue
    • Bulletin of the Korean Mathematical Society
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    • v.50 no.1
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    • pp.175-184
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    • 2013
  • In this paper, the reverse Bonnesen style inequalities for convex domain in the Euclidean plane $\mathbb{R}^2$ are investigated. The Minkowski mixed convex set of two convex sets K and L is studied and some new geometric inequalities are obtained. From these inequalities obtained, some isoperimetric deficit upper limits, that is, the reverse Bonnesen style inequalities for convex domain K are obtained. These isoperimetric deficit upper limits obtained are more fundamental than the known results of Bottema ([5]) and Pleijel ([22]).