Equilateral sets in the ${\ell }_1$ sum of Euclidean spaces

Abstract

Let $E^n$ denote the (real) $n$-dimensional Euclidean space. It is not known whether an equilateral set in the ${\ell }_1$ sum of $E^a$ and $E^b$, denoted here as $E^a{\oplus }_1{E}^b$, has maximum size at least $\dim (E^a{\oplus }_1{E}^b)+1=a+b+1$ for all pairs of $a$ and $b$. We show, via some explicit constructions of equilateral sets, that this holds for all $a⩽27$, as well as some other instances.