A problem of Kusner on equilateral sets

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 1<p<2 with d 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<p<2, where $\varepsilon_{p} > 0$ depends on p.