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

Abstract

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.