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 $a \leqslant 27$, as well as some other instances.