Geometry of multilinear forms on a normed space $\mathbb{R}^m$

Sung Guen Kim

doi:10.3842/umzh.v76i5.7476

Sung Guen Kim Department of Mathematics, Kyungpook National University, Daegu, South Korea

DOI: https://doi.org/10.3842/umzh.v76i5.7476

Keywords: multilinear forms, extreme points, exposed points.

Abstract

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)].

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