  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.
