• Journal of the Korean Mathematical Society
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• v.60 no.1
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• pp.91-113
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• 2023

The polytope of tristochastic tensors of degree three, the Latin polytope, has two kinds of extreme points. Those that are at a maximum distance from the barycenter of the polytope correspond to Latin squares. The remaining extreme points are said to be short. The aim of the paper is to determine the geometry of these short extreme points, as they relate to the Latin squares. The paper adapts the Latin square notion of an intercalate to yield the new concept of a cross-intercalate between two Latin squares. Cross-intercalates of pairs of orthogonal Latin squares of degree three are used to produce the short extreme points of the degree three Latin polytope. The pairs of orthogonal Latin squares fall into two classes, described as parallel and reversed, each forming an orbit under the isotopy group. In the inverse direction, we show that each short extreme point of the Latin polytope determines four pairs of orthogonal Latin squares, two parallel and two reversed.

• Communications of the Korean Mathematical Society
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• v.35 no.3
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• pp.799-807
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• 2020

We investigate extreme points of the unit ball of the space 𝓛(²𝒍_∞).

• Kyungpook Mathematical Journal
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• v.60 no.3
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• pp.485-505
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• 2020

We present a complete description of all the extreme points of the unit ball of 𝓛_s(²𝑙³_∞) which leads to a complete formula for ║f║ for every f ∈ 𝓛_s(²𝑙³_∞)^∗. We also show that $extB_{{\mathcal{L}}_s(^2l^3_{\infty})}{\subset}extB_{{\mathcal{L}}_s(^2l^n_{\infty})}$ for every n ≥ 4. Using the formula for ║f║ for every f ∈ 𝓛_s(²𝑙³_∞)^∗, we show that every extreme point of the unit ball of 𝓛_s(²𝑙³_∞) is exposed. We also characterize all the smooth points of the unit ball of 𝓛_s(²𝑙³_∞).

• Korean Journal of Mathematics
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• v.12 no.2
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• pp.107-115
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• 2004

In this paper, the new subclass denoted by $S_p({\alpha},{\beta},{\xi},{\gamma})$ of $p$-valent holomorphic functions has been introduced and investigate the several properties of the class $S_p({\alpha},{\beta},{\xi},{\gamma})$. In particular we have obtained integral representation for mappings in the class $S_p({\alpha},{\beta},{\xi},{\gamma})$) and determined closed convex hulls and their extreme points of the class $S_p({\alpha},{\beta},{\xi},{\gamma})$.

• Kyungpook Mathematical Journal
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• v.58 no.1
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• pp.47-54
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• 2018

We classify the extreme, exposed and smooth bilinear forms of the unit ball of the space of bilinear forms on $l^2_{\infty}$.

• Communications of the Korean Mathematical Society
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• v.32 no.4
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• pp.991-997
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• 2017

We classify the extreme, exposed and smooth symmetric 3-linear forms of the unit ball of ${\mathcal{L}}_s(^3l^2_{\infty})$, respectively.

• Communications of the Korean Mathematical Society
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• v.10 no.4
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• pp.875-880
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• 1995

Let $S_p(\alpha)$ denote the class of the Spirallike functions of order $\alpha, 0 < $\mid$\alpha$\mid$ < \frac{\pi}{2}$ Let $\Pi_N$ denote the subset of $S_p(\alpha)$ consisting of all products $z\Pi^N_{j=1}(1-u_j z)^{-mt_j}$ where $m = 1 + e^{-2i\alpha},$\mid$u_j$\mid$ = 1, t_j > 0$ for $j = 1, \cdots, N$ and $\sum^{N}_{j=1}{t_j = 1}$. In this paper we prove that extreme points of $S_p(\alpha)$ may be found which lie in $\Pi_N$ for some $N \geq 2$. We are let to conjecture that all exreme points of $S_p(\alpha)$ lie in $\Pi_N$ for somer $N \geq 1$ and that every such function is an extreme point.

• Kyungpook Mathematical Journal
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• v.57 no.2
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• pp.223-232
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• 2017

We classify the extreme and exposed symmetric bilinear forms of the unit ball of the space of symmetric bilinear forms on ${\mathbb{R}}^2$ with hexagonal norms. We also show that every extreme symmetric bilinear forms of the unit ball of the space of symmetric bilinear forms on ${\mathbb{R}}^2$ with hexagonal norms is exposed.

• Communications of the Korean Mathematical Society
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• v.34 no.2
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• pp.487-494
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• 2019

We classify all the extreme and exposed bilinear forms of the unit ball of ${\mathcal{L}}(^2l^3_{\infty})$ which leads to a complete formula of ${\parallel}f{\parallel}$ for every $f{\in}{\mathcal{L}}(^2l^3_{\infty})^*$. It follows from this formula that every extreme bilinear form of the unit ball of ${\mathcal{L}}(^2l^3_{\infty})$ is exposed.

• Journal of the Korean Mathematical Society
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• v.60 no.1
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• pp.213-225
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• 2023

For every n ≥ 2, let ℝⁿ_‖·‖ be Rn with a norm ‖·‖ such that its unit ball has finitely many extreme points more than 2n. We devote to the description of the sets of extreme and exposed points of the closed unit balls of 𝓛(²ℝⁿ_‖·‖) and 𝓛_𝒮(²ℝⁿ_‖·‖), where 𝓛(²ℝⁿ_‖·‖) is the space of bilinear forms on ℝⁿ_‖·‖, and 𝓛_𝒮(²ℝⁿ_‖·‖) is the subspace of 𝓛(²ℝⁿ_‖·‖) consisting of symmetric bilinear forms. Let 𝓕 = 𝓛(²ℝⁿ_‖·‖) or 𝓛_𝒮(²ℝⁿ_‖·‖). First we classify the extreme and exposed points of the closed unit ball of 𝓕. We also show that every extreme point of the closed unit ball of 𝓕 is exposed. It is shown that ext B_{𝓛𝒮(²ℝⁿ‖·‖)} = ext B_{𝓛(²ℝⁿ‖·‖)} ∩ 𝓛_𝒮(²ℝⁿ_‖·‖) and exp B_{𝓛𝒮(²ℝⁿ‖·‖)} = exp B_{𝓛(²ℝⁿ‖·‖)} ∩ 𝓛_𝒮(²ℝⁿ_‖·‖), which expand some results of [18, 23, 28, 29, 35, 38, 40, 41, 43].