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.