• 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.

• Kyungpook Mathematical Journal
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• v.54 no.3
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• pp.341-347
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• 2014

We classify the exposed symmetric bilinear forms of the unit ball of $\mathcal{L}_s(^2d_*(1,{\omega})^2)$.

• 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].

• 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_∞).