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