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arXiv · 1304.7033

Equilateral sets and a Schütte Theorem for the 4-norm

Konrad J. Swanepoel
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Abstract

A well-known theorem of Schütte (1963) gives a sharp lower bound for the ratio between the maximum distance and minimum distance between n+2 points in n-dimensional Euclidean space. In this brief note we adapt Bárány's elegant proof of this theorem to the space $\ell_4^n$. This gives a new proof that the largest cardinality of an equilateral set in $\ell_4^n$ is n+1, and gives a constructive bound for an interval $(4-ε_n,4+ε_n)$ of values of p close to 4 for which it is guaranteed that the largest cardinality of an equilateral set in $\ell_p^n$ is n+1.

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Konrad J. Swanepoel. 2014-11-10. Equilateral sets and a Schütte Theorem for the 4-norm. https://doi.org/10.4153/cmb-2013-031-0

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