Maximal equilateral sets

Abstract

A subset of a normed space X is called equilateral if the distance between any two points is the same. Let m(X) be the smallest possible size of an equilateral subset of X maximal with respect to inclusion. We first observe that Petty’s construction of a d-dimensional X of any finite dimension d ≥ 4 with m(X) = 4 can be generalised to give m(X ⊕1 R) = 4 for any X of dimension at least 2 which has a smooth point on its unit sphere. By a construction involving Hadamard matrices we then show that for any set Γ, m(ℓp(Γ)) is finite and bounded above by a function of p, for all 1 ≤ p < 2. Also, for all p ∈ [1, ∞) and d ∈ N there exists c = c(p, d) > 1 such that m(X) ≤ d + 1 for all d-dimensional X with Banach-Mazur distance less than c from ℓ d p. Using Brouwer’s fixed-point theorem we show that m(X) ≤ d+1 for all d-dimensional X with Banach-Mazur distance less than 3/2 from ℓ d∞. A graph-theoretical argument furthermore shows that m(ℓ d∞) = d + 1. The above results lead us to conjecture that m(X) ≤ 1 + dim X for all finite-dimensional normed spaces X.