Equilateral sets and a Schütte theorem for the 4-norm (Q2925383)

Let \(X\) be a normed linear space. A set \(S \subset X\) is said to be equilateral if there is a \(\lambda >0\) such that for any distinct \(x,y \in S\), \(\|x-y\|\) is constant. In a finite-dimensional space, the largest cardinality of an equilateral set is denoted by \(e(X)\). This interesting paper investigates this question when \(X = \ell^n_p\). When \(p = 4\), this number is known to be \(n+1\). The paper exhibits another neighborhood of \(4\), where \(e(X) = n+1\) (see the article by \textit{C. Smyth} [``Equilateral sets in \(\ell^d_p\)'', in: Thirty essays on geometric graph theory. Berlin: Springer. 483--487 (2013; Zbl 1276.52020)]).