• Title, Summary, Keyword: convex domain

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DIFFERENTIABILITY OF QUASI-HOMOGENEOUS CONVEX AFFINE DOMAINS

  • JO KYEONGHEE
    • Journal of the Korean Mathematical Society
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    • v.42 no.3
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    • pp.485-498
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    • 2005
  • In this article we show that every quasi-homogeneous convex affine domain whose boundary is everywhere differentiable except possibly at a finite number of points is either homogeneous or covers a compact affine manifold. Actually we show that such a domain must be a non-elliptic strictly convex cone if it is not homogeneous.

NONEXISTENCE OF H-CONVEX CUSPIDAL STANDARD FUNDAMENTAL DOMAIN

  • Yayenie, Omer
    • Bulletin of the Korean Mathematical Society
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    • v.46 no.5
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    • pp.823-833
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    • 2009
  • It is well-known that if a convex hyperbolic polygon is constructed as a fundamental domain for a subgroup of the modular group, then its translates by the group elements form a locally finite tessellation and its side-pairing transformations form a system of generators for the group. Such hyperbolically convex polygons can be obtained by using Dirichlet's and Ford's polygon constructions. Another method of obtaining a fundamental domain for subgroups of the modular group is through the use of a right coset decomposition and we call such domains standard fundamental domains. In this paper we give subgroups of the modular group which do not have hyperbolically convex standard fundamental domain containing only inequivalent cusps.

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]).

ASYMPTOTIC FOLIATIONS OF QUASI-HOMOGENEOUS CONVEX AFFINE DOMAINS

  • Jo, Kyeonghee
    • Communications of the Korean Mathematical Society
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    • v.32 no.1
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    • pp.165-173
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    • 2017
  • In this paper, we prove that the automorphism group of a quasi-homogeneous properly convex affine domain in ${\mathbb{R}_n}$ acts transitively on the set of all the extreme points of the domain. This set is equal to the set of all the asymptotic cone points coming from the asymptotic foliation of the domain and thus it is a homogeneous submanifold of ${\mathbb{R}_n}$.

COEFFICIENT BOUNDS FOR CLOSE-TO-CONVEX FUNCTIONS ASSOCIATED WITH VERTICAL STRIP DOMAIN

  • Bulut, Serap
    • Communications of the Korean Mathematical Society
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    • v.35 no.3
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    • pp.789-797
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    • 2020
  • By considering a certain univalent function in the open unit disk ��, that maps �� onto a strip domain, we introduce a new class of analytic and close-to-convex functions by means of a certain non-homogeneous Cauchy-Euler-type differential equation. We determine the coefficient bounds for functions in this new class. Relevant connections of some of the results obtained with those in earlier works are also provided.

$C^\infty$ EXTENSIONS OF HOLOMORPHIC FUNCTIONS FROM SUBVARIETIES OF A CONVEX DOMAIN

  • Cho, Hong-Rae
    • Bulletin of the Korean Mathematical Society
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    • v.38 no.3
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    • pp.487-493
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    • 2001
  • $Let \Omega$ be a bounded convex domain in C^n$ with smooth boundary. Let M be a subvariety of $\Omega$ which intersects $\partial$$\Omega$ transversally. Suppose that $\Omega$ is totally convex at any point of $\partial$M in the complex tangential directions.For f $\epsilon$O(M)$\bigcap$/TEX>$C^{\infty}$($\overline{M}$/TEX>), there exists F $\epsilon$ o ($\Omega$))$\bigcap$/TEX>$C^{\infty}$($\overline{\Omega}$/TEX>) such that F(z) = f(z) for z $\epsilon$ M.

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