Large equilateral sets in subspaces of ${\mathrm{\ell }}_{\mathrm{\infty }}^n$ of small codimension

For fixed $k$ we prove exponential lower bounds on the equilateral number of subspaces of ${\mathrm{\ell }}_{\mathrm{\infty }}^n$ of codimension $k$. In particular, we show that if the unit ball of a normed space of dimension $n$ is a centrally symmetric polytope with at most $\frac{4n}{3}-o(n)$ pairs of facets, then it has an equilateral set of cardinality at least $n+1$. These include subspaces of codimension $2$ of ${\mathrm{\ell }}_{\mathrm{\infty }}^{n+2}$ for $n\ge 9$ and of codimension $3$ of ${\mathrm{\ell }}_{\mathrm{\infty }}^{n+3}$ for $n\ge 15$.