ACTA issues

Geometry of multilinear forms on $\mathbb {R}^m$ with a certain norm

Sung Guen Kim

Acta Sci. Math. (Szeged) 87:1-2(2021), 207-219

Abstract. For every $m\geq 2$, let $\mathbb {R}^m_{\|\cdot \|}$ be $\mathbb {R}^m$ with a norm $\|\cdot \|$ such that $|{ext} B_{\mathbb {R}^m_{\|\cdot \|}}|=2m$. For every $n\geq 2,$ we devote ourselves to 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 ${\mathcal L}(^n\mathbb {R}^m_{\|\cdot \|})$ consisting of 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 classify the extreme and exposed points of the closed unit ball of ${\mathcal F}$. We obtain $\big |{ext}B_{{\mathcal L}(^n\mathbb {R}^m_{\|\cdot \|})}\big |=2^{(m^n)}$ and $\big | {ext}B_{{\mathcal L}_s(^n\mathbb {R}^m_{\|\cdot \|})}\big |=2^{o{dim}({\mathcal L}_s(^n\mathbb {R}^m_{\||\cdot \||}))}$. We also show that every extreme point of the closed unit ball of ${\mathcal F}$ is exposed. It is shown that ${ext}B_{{\mathcal L}_s(^n\mathbb {R}^m_{\|\cdot \|})}={ext}B_{{\mathcal L}(^n\mathbb {R}^m_{\|\cdot \|})}\cap {\mathcal L}_s(^n\mathbb {R}^m_{\|\cdot \|})$ and $ o{exp}B_{{\mathcal L}_s(^n\mathbb {R}^m_{\|\cdot \|})}= o{exp}B_{{\mathcal L}(^n\mathbb {R}^m_{\|\cdot \|})}\cap {\mathcal L}_s(^n\mathbb {R}^m_{\|\cdot \|})$.

DOI: 10.14232/actasm-020-824-2

AMS Subject Classification (1991): 46A22

Keyword(s): multilinear forms, symmetric multilinear forms, extreme points, exposed points

received 24.8.2020, revised 8.12.2020, accepted 14.12.2020. (Registered under 824/2020.)