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Equilateral Sets and a Schütte Theorem for the $4$-norm

  Published:2013-12-04
 Printed: Sep 2014
  • Konrad J. Swanepoel,
    Department of Mathematics, London School of Economics and Political Science, Houghton Street, London WC2A 2AE, United Kingdom
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Abstract

A well-known theorem of Schütte (1963) gives a sharp lower bound for the ratio of the maximum and minimum distances between $n+2$ points in $n$-dimensional Euclidean space. In this note we adapt Bárány's elegant proof (1994) 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 $(4-\varepsilon_n,4+\varepsilon_n)$ of values of $p$ close to $4$ for which it is known that the largest cardinality of an equilateral set in $\ell_p^n$ is $n+1$.
MSC Classifications: 46B20, 52A21, 52C17 show english descriptions Geometry and structure of normed linear spaces
Finite-dimensional Banach spaces (including special norms, zonoids, etc.) [See also 46Bxx]
Packing and covering in $n$ dimensions [See also 05B40, 11H31]
46B20 - Geometry and structure of normed linear spaces
52A21 - Finite-dimensional Banach spaces (including special norms, zonoids, etc.) [See also 46Bxx]
52C17 - Packing and covering in $n$ dimensions [See also 05B40, 11H31]
 

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