Open Access
2013/2014 Equilateral Weights on the Unit Ball of ℝ n
Emmanuel Chetcuti, Joseph Muscat
Real Anal. Exchange 40(1): 37-52 (2013/2014).


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.


Download Citation

Emmanuel Chetcuti. Joseph Muscat. "Equilateral Weights on the Unit Ball of ℝ n ." Real Anal. Exchange 40 (1) 37 - 52, 2013/2014.


Published: 2013/2014
First available in Project Euclid: 1 July 2015

zbMATH: 06848822
MathSciNet: MR3365389

Primary: 26A03 , 26A04
Secondary: 26A05

Keywords: Equilateral set , Equilateral weight

Rights: Copyright © 2015 Michigan State University Press

Vol.40 • No. 1 • 2013/2014
Back to Top