On Nearly Equilateral Simplices and Nearly l∞ Spaces

Abstract

AbstractBy 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 d(X, ) ≤ 2, where stands for ℝn endowed with the norm ║(x1, … , xn)║∞ := max﹛|x1|, … , |xn|﹜. Then every metric space (S, ρ) of cardinality n + 1 with norm ρ satisfying the condition maxD/minD ≤ 2/ d(X, ) for D := ﹛ρ(a, b) : a, b ∈ S, a ≠ b﹜ can be isometrically embedded into X.