Equilateral weights on the unit ball of \(\mathbb{R}^n\) (Q1704416)

An equilateral set (a regular simplex) in a metric space is a set for which the distance of any pair of distinct points is the same number. It is called standard, if this number is 1. For a fixed \(W \in \mathbb{R}\), an equilateral weight if the sum of its values at points of an arbitrary standard equilateral set is \(W\). The paper studies properties of equilateral sets and equilateral weights in unit balls of \(n\)-dimensional Euclidean spaces. It is shown that for dimensions over 2 every equilateral weight on a unit ball is constant.