Equilateral sets and a Sch\"utte Theorem for the 4-norm

TL;DR

This paper extends Sch"utte's theorem to the space ^n, providing a new proof that the maximum size of equilateral sets is n+1 and establishing bounds for p near 4 where this holds.

Contribution

It adapts Be1re1ny's proof of Schfctte's theorem to ^n, offering a new proof and bounds for equilateral set sizes in ^n and ^n-like spaces.

Findings

Largest equilateral set size in ^n is n+1.

Constructive bounds for p near 4 where the maximum set size is n+1.

New proof of Schfctte's theorem adapted to ^n.

Abstract

A well-known theorem of Sch\"utte (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 B\'ar\'any'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.