Geometry of multilinear forms on a normed space $\mathbb{R}^m$
UDC 514.1 For every $m\geq 2,$ let $\mathbb{R}^m_{\|\cdot\|}$ be $\mathbb{R}^m$ with a norm $\|\cdot\|$ such that its unit ball has finitely many extreme points. For every $n\geq2,$ we focus our attention on the description of the sets of extreme and exposed points of the closed unit ba...
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| Datum: | 2024 |
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Institute of Mathematics, NAS of Ukraine
2024
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860512674744369152 |
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| author | Kim, Sung Guen Kim, Sung Guen |
| author_facet | Kim, Sung Guen Kim, Sung Guen |
| author_sort | Kim, Sung Guen |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2024-07-15T03:05:03Z |
| description | UDC 514.1
For every $m\geq 2,$ let $\mathbb{R}^m_{\|\cdot\|}$ be $\mathbb{R}^m$ with a norm $\|\cdot\|$ such that its unit ball has finitely many extreme points. For every $n\geq2,$ we focus our attention on the description of the sets of extreme and exposed points of the closed unit balls of ${\mathcal L}(^n\mathbb{R}^m_{\|\cdot\|})$ and ${\mathcal L}_s(^n\mathbb{R}^m_{\|\cdot\|})$, where ${\mathcal L}(^n\mathbb{R}^m_{\|\cdot\|})$ is the space of $n$-linear forms on $\mathbb{R}^m_{\|\cdot\|}$ and ${\mathcal L}_s(^n\mathbb{R}^m_{\|\cdot\|})$ is the subspace of ${\mathcal L}(^n\mathbb{R}^m_{\|\cdot\|})$ formed by symmetric $n$-linear forms. Let ${\mathcal F}={\mathcal L}(^n\mathbb{R}^m_{\|\cdot\|})$ or ${\mathcal L}_s(^n\mathbb{R}^m_{\|\cdot\|}).$  First, we show that the number of extreme points of the unit ball of $\mathbb{R}^m_{\|\cdot\|}$ is greater than $2m.$ By using this fact, we classify the extreme and exposed points of the closed unit ball of ${\mathcal F},$ respectively.  It is shown that every extreme point of the closed unit ball of ${\mathcal F}$ is exposed. We obtain the results of [Studia Sci. Math. Hungar., 57, No. 3, 267–283 (2020)] and extend the results of [Acta Sci. Math. Szedged, 87, No. 1-2, 233–245 (2021) and J. Korean Math., Soc., 60, No. 1-2, 213–225 (2023)]. |
| doi_str_mv | 10.3842/umzh.v76i5.7476 |
| first_indexed | 2026-03-24T03:32:33Z |
| format | Article |
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| id | umjimathkievua-article-7476 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
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| language | English |
| last_indexed | 2026-03-24T03:32:33Z |
| publishDate | 2024 |
| publisher | Institute of Mathematics, NAS of Ukraine |
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| spelling | umjimathkievua-article-74762024-07-15T03:05:03Z Geometry of multilinear forms on a normed space $\mathbb{R}^m$ Geometry of multilinear forms on a normed space $\mathbb{R}^m$ Kim, Sung Guen Kim, Sung Guen multilinear forms, extreme points, exposed points. UDC 514.1 For every $m\geq 2,$ let $\mathbb{R}^m_{\|\cdot\|}$ be $\mathbb{R}^m$ with a norm $\|\cdot\|$ such that its unit ball has finitely many extreme points. For every $n\geq2,$ we focus our attention on the description of the sets of extreme and exposed points of the closed unit balls of ${\mathcal L}(^n\mathbb{R}^m_{\|\cdot\|})$ and ${\mathcal L}_s(^n\mathbb{R}^m_{\|\cdot\|})$, where ${\mathcal L}(^n\mathbb{R}^m_{\|\cdot\|})$ is the space of $n$-linear forms on $\mathbb{R}^m_{\|\cdot\|}$ and ${\mathcal L}_s(^n\mathbb{R}^m_{\|\cdot\|})$ is the subspace of ${\mathcal L}(^n\mathbb{R}^m_{\|\cdot\|})$ formed by symmetric $n$-linear forms. Let ${\mathcal F}={\mathcal L}(^n\mathbb{R}^m_{\|\cdot\|})$ or ${\mathcal L}_s(^n\mathbb{R}^m_{\|\cdot\|}).$  First, we show that the number of extreme points of the unit ball of $\mathbb{R}^m_{\|\cdot\|}$ is greater than $2m.$ By using this fact, we classify the extreme and exposed points of the closed unit ball of ${\mathcal F},$ respectively.  It is shown that every extreme point of the closed unit ball of ${\mathcal F}$ is exposed. We obtain the results of [Studia Sci. Math. Hungar., 57, No. 3, 267–283 (2020)] and extend the results of [Acta Sci. Math. Szedged, 87, No. 1-2, 233–245 (2021) and J. Korean Math., Soc., 60, No. 1-2, 213–225 (2023)]. УДК 514.1 Геометрія багатолінійних форм на нормованому просторі $\mathbb{R}^m$ Нехай $\mathbb{R}^m_{\|\cdot\|}$ для кожного $m\geq 2$ — це $\mathbb{R}^m$ з нормою $\|\cdot\|$ такою, що її одинична куля має скінченну кількість екстремальних точок.  Для кожного $n\geq2$ ми приділимо увагу опису множин екстремальних та відкритих точок замкнених одиничних куль в ${\mathcal L}(^n\mathbb{R}^m_{\|\cdot\|})$ і ${\mathcal L}_s(^n\mathbb{R}^m_{\|\cdot\|})$, де ${\mathcal L}(^n\mathbb{R}^m_{\|\cdot\|})$ — простір $n$-лінійних форм на $\mathbb{R}^m_{\|\cdot\|},$ а ${\mathcal L}_s(^n\mathbb{R}^m_{\|\cdot\|})$ — підпростір ${\mathcal L}(^n\mathbb{R}^m_{\|\cdot\|})$, що складається з симетричних $n$-лінійних форм. Нехай ${\mathcal F}={\mathcal L}(^n\mathbb{R}^m_{\|\cdot\|})$ або ${\mathcal L}_s(^n\mathbb{R}^ m_{\|\cdot\|}).$ Спочатку ми показуємо, що кількість екстремальних точок одиничної кулі $\mathbb{R}^m_{\|\cdot\|}$ більша ніж $2m.$ Використовуючи цей факт, ми класифікуємо екстремальні  та відкриті точки замкненої одиничної кулі в ${\mathcal F}$ відповідно. Показано, що кожна екстремальна точка замкненої одиничної кулі  ${\mathcal F}$ є відкритою. Отримано результати роботи [Studia Sci. Math. Hungar., 57, No. 3, 267–283 (2020)] та розширено результати з [Acta Sci. Math. Szedged, 87, No. 1-2, 233–245 (2021) and J. Korean Math., Soc., 60, No. 1-2, 213–225 (2023)]. Institute of Mathematics, NAS of Ukraine 2024-07-03 Article Article https://umj.imath.kiev.ua/index.php/umj/article/view/7476 10.3842/umzh.v76i5.7476 Ukrains’kyi Matematychnyi Zhurnal; Vol. 76 No. 6 (2024); 855–863 Український математичний журнал; Том 76 № 6 (2024); 855–863 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/7476/10032 Copyright (c) 2024 Sung Guen Kim |
| spellingShingle | Kim, Sung Guen Kim, Sung Guen Geometry of multilinear forms on a normed space $\mathbb{R}^m$ |
| title | Geometry of multilinear forms on a normed space $\mathbb{R}^m$ |
| title_alt | Geometry of multilinear forms on a normed space $\mathbb{R}^m$ |
| title_full | Geometry of multilinear forms on a normed space $\mathbb{R}^m$ |
| title_fullStr | Geometry of multilinear forms on a normed space $\mathbb{R}^m$ |
| title_full_unstemmed | Geometry of multilinear forms on a normed space $\mathbb{R}^m$ |
| title_short | Geometry of multilinear forms on a normed space $\mathbb{R}^m$ |
| title_sort | geometry of multilinear forms on a normed space $\mathbb{r}^m$ |
| topic_facet | multilinear forms extreme points exposed points. |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/7476 |
| work_keys_str_mv | AT kimsungguen geometryofmultilinearformsonanormedspacemathbbrm AT kimsungguen geometryofmultilinearformsonanormedspacemathbbrm |