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# On Nearly Equilateral Simplices and Nearly l∞ Spaces

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Published:2010-05-11
Printed: Sep 2010
• Gennadiy Averkov,
Institute for Mathematical Optimization, Faculty of Mathematics, University of Magdeburg, Magdeburg, Germany
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## Abstract

By $\textrm{d}(X,Y)$ we denote the (multiplicative) Banach--Mazur distance between two normed spaces $X$ and $Y.$ Let $X$ be an $n$-dimensional normed space with $\textrm{d}(X,\ell_\infty^n) \le 2,$ where $\ell_\infty^n$ stands for $\mathbb{R}^n$ endowed with the norm $\|(x_1,\dots,x_n)\|_\infty := \max \{|x_1|,\dots, |x_n| \}.$ Then every metric space $(S,\rho)$ of cardinality $n+1$ with norm $\rho$ satisfying the condition $\max D / \min D \le 2/ \textrm{d}(X,\ell_\infty^n)$ for $D:=\{ \rho(a,b) : a, b \in S, \ a \ne b\}$ can be isometrically embedded into $X.$
 MSC Classifications: 52A21 - Finite-dimensional Banach spaces (including special norms, zonoids, etc.) [See also 46Bxx] 51F99 - None of the above, but in this section 52C99 - None of the above, but in this section

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