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A problem of Kusner on equilateral sets

Authors:	Email author" target="_blank">K?J?Swanepoel Email author

Institution:	(1) Department of Mathematics, Applied Mathematics and Astronomy, University of South Africa, P.O. Box 392, 0003 Pretoria, South Africa

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. Received: 5 May 2003

Keywords:	Primary 52C10 Secondary 52A21 46B20
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