Equilateral Sets in ${\mathcal{l}}_{p}^{d}$

Abstract

If X is a Minkowski space, i.e., a finite-dimensional real normed space, then $S \subset X$ is an equilateral set if all pairs of points of S determine the same distance with respect to the norm. Kusner conjectured that $e({\mathcal{l}}_{p}^{d}) = d + 1$ for $1 < p < \infty$ and $e({\mathcal{l}}_{1}^{d}) = 2d$ [6]. Using a technique combining linear algebra and approximation theory, we prove that for all $1 < p < \infty$, there exists a constant C _p > 0 such that $e({\mathcal{l}}_{p}^{d}) \leq {C}_{p}{d}^{1+2/(p-1)}$.