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Equilateral weights on the unit ball of $\mathbb R^n$

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 \cite{Gl}, we define an equilateral weight on a metric space $X$ to be a function $f:X\to \mathbb R$ 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.