An equilateral set \(A:=\{a_i: i\in I\}\) in a normed space is one for which \(\| a_i - a_j\| = \delta_{ij}b\) (the scale factor \(b > 0\) is sometimes useful). The vertices of the unit ball in \(\ell^n_{\infty}\) and in \(\ell^n_1\) form equilateral sets (with \(b=2\)) of cardinality \(2^n\) and \(2n\), respectively. Let \(e(\ell^n_p)\) denote the cardinality of a maximal equilateral set in \(\ell^n_p\). It is well-known that \(e(\ell^n_2) = n+1\). It has been conjectured that \(e(\ell^n_1) = 2n\) and \(e(\ell^n_p) = n+1\) for \(p\neq 1,\infty\). Only the trivial upper bound of \(2^n - 1\) for \(e(\ell^n_1)\) was previously known, but for \(p\) an even integer it is known that \(e(\ell^n_p) \leq 1 + (p-1)n\).
Here the authors achieve a major breakthrough and show that there is an absolute constant \(c\) such that \(e(\ell^n_1) \leq cn\log n\) and, more generally, for each odd integer \(p \geq 1\) a constant \(c_p\) such that \(e(\ell^n_p) \leq c_pn\log n.\) The bulk of the paper (about 9 pages) is taken up with the details of the proof of the \(\ell^n_1\) result.
A long-standing open question is whether, in every normed space \(X\) of dimension \(n\), \(e(X) \geq n+1\).