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 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 \(f:X\longrightarrow \mathbb{R}\) such that \(\sum_{i\in I}f(x_i)=W\) for every maximal standard equilateral set \(\{x_i:i\in I\}\) in \(X\), where \(W\in\mathbb{R}\) is the weight of \(f\). In this paper, we characterize the equilateral weights associated with the unit ball \(B^n\) of \(\mathbb{R}^n\) as follows: For \(n\ge 2\), every equilateral weight on \(B^n\) is constant.