EQUILATERAL WEIGHTS ON THE UNIT BALL OF R-n

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 sigma(i is an element of I )f(x(i)) = W for every maximal standard equilateral set {x(i) : i is an element of I} in X, where W is an element of R is the weight of f. In this paper, we characterize the equilateral weights associated with the unit ball B-n of R-n as follows: For n > >= 2, every equilateral weight on B-n is constant.