This is an announcement for the paper "Equilateral sets and a Sch\"utte Theorem for the 4-norm" by Konrad J. Swanepoel. 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. Archive classification: math.MG math.FA Mathematics Subject Classification: Primary 46B20, Secondary 52A21, 52C17 Remarks: 5 pages Submitted from: konrad.swanepoel@gmail.com The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1304.7033 or http://arXiv.org/abs/1304.7033