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Maximal equilateral sets

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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 show that m(X\oplus_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 both m(\ell_p) and m(\ell_p^d) are finite and bounded above by a function of p, for all 1 <= p < 2. Also, for all p in [1,\infty) and natural numbers d 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 \ell_p^d. 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 \ell_\infty^d. A graph-theoretical argument furthermore shows that m(\ell_\infty^d)=d+1. The above results lead us to conjecture that m(X) <= 1+\dim X.

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