Equilateral sets and a Schütte Theorem for the 4-norm

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 $(4-{\epsilon }_n,4+{\epsilon }_n)$ 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.

7 pages. Some technical details added

References in corpus (1)

• A problem of Kusner on equilateral sets