Large equilateral sets in subspaces of ℓ∞n of small codimension

For fixed k we prove exponential lower bounds on the equilateral number of subspaces of ℓ∞n of codimension k. In particular, we show that subspaces of codimension 2 of ℓ∞n+2 and subspaces of codimension 3 of ℓ∞n+3 have an equilateral set of cardinality n+ 1 if n≥ 7 and 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 4 n/ 3 - o(n) pairs of facets.