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For fixed  k  we prove exponential lower bounds on the equilateral number of subspaces of    $$\ell _{\infty }^n$$  ℓ ∞ n of codimension  k . In particular, we show that subspaces of codimension 2 of    $$\ell _{\infty }^{n+2}$$  ℓ ∞ n + 2 and subspaces of codimension 3 of    $$\ell _{\infty }^{n+3}$$  ℓ ∞ n + 3 have an equilateral set of cardinality    $$n+1$$  n + 1 if    $$n\ge 7$$  n ≥ 7 and    $$n\ge 12$$  n ≥ 12 respectively. Moreover, the same is true for every normed space of dimension  n , whose unit ball is a centrally symmetric polytope with at most    $${4n}/{3}-o(n)$$  4 n / 3 - o ( n ) pairs of facets.

Large Equilateral Sets in Subspaces of $$\ell _\infty ^n$$ of Small Codimension