Archiv der Mathematik

, Volume 83, Issue 2, pp 164–170

A problem of Kusner on equilateral sets

Original paper

DOI: 10.1007/s00013-003-4840-8

Cite this article as:
Swanepoel, K.J. Arch. Math. (2004) 83: 164. doi:10.1007/s00013-003-4840-8

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.

Mathematics Subject Classification (2000):

Primary 52C10Secondary 52A2146B20.

Copyright information

© Birkhäuser-Verlag 2004

Authors and Affiliations

  1. 1.Department of Mathematics, Applied Mathematics and AstronomyUniversity of South AfricaPretoriaSouth Africa