Equilateral weights on the unit ball of ℝ ⁿ

Abstract

An equilateral set (or regular simplex) in a metric space X , is a set A such that the distance between any pair of distinct members of A is a constant. An equilateral set is standard if the distance between distinct members is equal to 1 . Motivated by the notion of frame-functions, as introduced and characterized by Gleason in [6], we define an equilateral weight on a metric space X to be a function f:X→R such that ∑i∈If(xi)=W , for every maximal standard equilateral set {xi:i∈I} in X , where W∈R is the weight of f . In this paper we characterize the equilateral weights associated with the unit ball Bn of Rn as follows: For n≥2 , every equilateral weight on Bn is constant.