On Nearly Equilateral Simplices and Nearly l_∞ Spaces

http://dx.doi.org/10.4153/CMB-2010-055-1
Canad. Math. Bull. 53(2010), 394-397

  Published:2010-05-11
 Printed: Sep 2010

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 $X.$