Maximal equilateral sets

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 -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(ℓ p (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 -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 -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+dimX for all finite-normed spaces X .