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