• Title, Summary, Keyword: the isoperimetric inequality

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ISOPERIMETRIC INEQUALITY IN α-PLANE

  • Kim, Min Seong;Ko, Il Seog;Kim, Byung Hak
    • Journal of applied mathematics & informatics
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    • v.31 no.1_2
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    • pp.79-86
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    • 2013
  • Taxicab plane geometry and Cinese-Checker plane geometry are non-Euclidean and more practical notion than Euclidean geometry in the real world. The ${\alpha}$-distance is a generalization of the Taxicab distance and Chinese-Checker distance. It was first introduced by Songlin Tian in 2005, and generalized to n-dimensional space by Ozcan Gelisgen in 2006. In this paper, we studied the isoperimetric inequality in ${\alpha}$-plane.

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.

HYPERBOLIC NOTIONS ON A PLANAR GRAPH OF BOUNDED FACE DEGREE

  • OH, BYUNG-GEUN
    • Bulletin of the Korean Mathematical Society
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    • v.52 no.4
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    • pp.1305-1319
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    • 2015
  • We study the relations between strong isoperimetric inequalities and Gromov hyperbolicity on planar graphs, and give an alternative proof for the following statement: if a planar graph of bounded face degree satisfies a strong isoperimetric inequality, then it is Gromov hyperbolic. This theorem was formerly proved in the author's paper from 2014 [12] using combinatorial methods, while geometric approach is used in the present paper.

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]).

RELATIVE ISOPERIMETRIC INEQUALITY FOR MINIMAL SUBMANIFOLDS IN SPACE FORMS

  • Seo, Keomkyo
    • Korean Journal of Mathematics
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    • v.18 no.2
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    • pp.195-200
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    • 2010
  • Let C be a closed convex set in ${\mathbb{S}}^m$ or ${\mathbb{H}}^m$. Assume that ${\Sigma}$ is an n-dimensional compact minimal submanifold outside C such that ${\Sigma}$ is orthogonal to ${\partial}C$ along ${\partial}{\Sigma}{\cap}{\partial}C$ and ${\partial}{\Sigma}$ lies on a geodesic sphere centered at a fixed point $p{\in}{\partial}{\Sigma}{\cap}{\partial}C$ and that r is the distance in ${\mathbb{S}}^m$ or ${\mathbb{H}}^m$ from p. We make use of a modified volume $M_p({\Sigma})$ of ${\Sigma}$ and obtain a sharp relative isoperimetric inequality $$\frac{1}{2}n^n{\omega}_nM_p({\Sigma})^{n-1}{\leq}Vol({\partial}{\Sigma}{\sim}{\partial}C)^n$$, where ${\omega}_n$ is the volume of a unit ball in ${\mathbb{R}}^n$ Equality holds if and only if ${\Sigma}$ is a totally geodesic half ball centered at p.

MIXED CHORD-INTEGRALS OF STAR BODIES

  • Fenghong, Lu
    • Journal of the Korean Mathematical Society
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    • v.47 no.2
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    • pp.277-288
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    • 2010
  • The mixed chord-integrals are defined. The Fenchel-Aleksandrov inequality and a general isoperimetric inequality for the mixed chordintegrals are established. Furthermore, the dual general Bieberbach inequality is presented. As an application of the dual form, a Brunn-Minkowski type inequality for mixed intersection bodies is given.

MIXED BRIGHTNESS-INTEGRALS OF CONVEX BODIES

  • Li, Ni;Zhu, Baocheng
    • Journal of the Korean Mathematical Society
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    • v.47 no.5
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    • pp.935-945
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    • 2010
  • The mixed width-integrals of convex bodies are defined by E. Lutwak. In this paper, the mixed brightness-integrals of convex bodies are defined. An inequality is established for the mixed brightness-integrals analogous to the Fenchel-Aleksandrov inequality for the mixed volumes. An isoperimetric inequality (involving the mixed brightness-integrals) is presented which generalizes an inequality recently obtained by Chakerian and Heil. Strengthened version of this general inequality is obtained by introducing indexed mixed brightness-integrals.

ROUGH ISOMETRY AND HARNACK INEQUALITY

  • Park, Hyeong-In;Lee, Yong-Hah
    • Journal of the Korean Mathematical Society
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    • v.33 no.2
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    • pp.455-468
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    • 1996
  • Certain analytic behavior of geometric objects defined on a Riemannian manifold depends on some very crude properties of the manifold. Some of those crude invariants are the volume growth rate, isoperimetric constants, and the likes. However, these crude invariants sometimes exercise surprising control over the analytic behavior.

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VOLUME INEQUALITIES FOR THE Lp-SINE TRANSFORM OF ISOTROPIC MEASURES

  • Guo, LuJun;Leng, Gangsong
    • Bulletin of the Korean Mathematical Society
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    • v.52 no.3
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    • pp.837-849
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    • 2015
  • For $p{\geq}1$, sharp isoperimetric inequalities for the $L_p$-sine transform of isotropic measures are established. The corresponding reverse inequalities are obtained in an asymptotically optimal form. As applications of our main results, we present volume inequalities for convex bodies which are in $L_p$ surface isotropic position.