Equilateral sets in the \(\ell_1\) sum of Euclidean spaces (Q2292906)

A subset of a metric space is called equilateral if its elements are all equidistant from one another. A longstanding conjecture is that any \(n\)-dimensional Banach space contains an equilateral set with cardinality at least \(n+1\). This is examined here in the context of the \(\ell_1\) sum of an \(a\)-dimensional Euclidean space and a \(b\)-dimensional Euclidean space. Without loss of generality, one may suppose \(a\le b\). The conjecture is established in case either \(b-1\), \(b\) or \(b+1\) is a multiple of \(a+1\), or if a certain inequality holds. For fixed \(a\), it can be shown that this inequality is satisfied for all but at most finitely many values of~\(b\). The smallest values of \((a,b)\) for which the conjecture remains open are \((28,40)\) and \((29,39)\).