, Volume 83, Issue 2, pp 164-170

A problem of Kusner on equilateral sets

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Abstract.

R. B. Kusner [R. Guy, Amer. Math. Monthly 90, 196-199 (1983)] asked whether a set of vectors in \( {\mathbb R}^{d} \) such that the \( \ell_p \) distance between any pair is 1, has cardinality at most d + 1. We show that this is true for p = 4 and any \( d \geq 1 \) , and false for all 1d sufficiently large, depending on p. More generally we show that the maximum cardinality is at most \( (2\lceil p/4\rceil-1)d+1 \) if p is an even integer, and at least \( (1 + \varepsilon_p)d \) if 1 \( \varepsilon_{p} > 0 \) depends on p.

Received: 5 May 2003