Abstract view
On Nearly Equilateral Simplices and Nearly l_{∞} Spaces


Published:20100511
Printed: Sep 2010
Gennadiy Averkov,
Institute for Mathematical Optimization, Faculty of Mathematics, University of Magdeburg, Magdeburg, Germany
Abstract
By $\textrm{d}(X,Y)$ we denote the (multiplicative) BanachMazur 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, 51F99, 52C99 show english descriptions
Finitedimensional Banach spaces (including special norms, zonoids, etc.) [See also 46Bxx] None of the above, but in this section None of the above, but in this section
52A21  Finitedimensional 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
