Abstract view

# Equilateral sets and a Schütte Theorem for the $4$-norm

Published:2013-12-04

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 - 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]