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On Nearly Equilateral Simplices and Nearly l∞ Spaces

Published online by Cambridge University Press:  20 November 2018

Gennadiy Averkov
Gennadiy Averkov*
Affiliation:
Institute for Mathematical Optimization, Faculty of Mathematics, University of Magdeburg, Magdeburg, Germany e-mail: gennadiy.averkov@ovgu.de
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Abstract

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By $\text{d(X,Y)}$ we denote the (multiplicative) Banach–Mazur distance between two normed spaces $X$ and $Y$. Let $X$ be an $n$-dimensional normed space with $\text{d(X,}\,l_{\infty }^{n}\text{)}\,\le \,\text{2}$, where $l_{\infty }^{n}$ stands for ${{\mathbb{R}}^{n}}$ endowed with the norm $\parallel ({{x}_{1}},\,.\,.\,.\,,\,{{x}_{n}}){{\parallel }_{\infty }}\,:=\,\max \{|{{x}_{1}}|,\,.\,.\,.\,,\,|{{x}_{n}}|\}$. Then every metric space $(S,\,\rho )$ of cardinality $n+1$ with norm $\rho $ satisfying the condition $\max D/\min D\,\le \,2/\,\text{d(}X,\,l_{\infty }^{n}\text{)}$ for $D\,:=\,\{\rho (a,\,b)\,:\,a,\,b\,\in \,S,\,a\,\ne \,b\}$ can be isometrically embedded into $X$.

Keywords

52A2151F9952C99
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 53 , Issue 3 , 01 September 2010 , pp. 394 - 397
Copyright
Copyright © Canadian Mathematical Society 2010

References

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