Equilateral Sets and a Schütte Theorem for the 4-norm

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 . This gives a new proof that the largest cardinality of an equilateral set in is n + 1 and gives a constructive bound for an interval (4–εn, 4 + εn) of values of p close to 4 for which it is known that the largest cardinality of an equilateral set in is n + 1.