Acute sets of exponentially optimal size

Abstract

We present a simple construction of an acute set of size $2^{d-1}+1$ in ${\mathbb{R}}^d$ for any dimension $d$. That is, we explicitly give $2^{d-1}+1$ points in the $d$-dimensional Euclidean space with the property that any three points form an acute triangle. It is known that the maximal number of such points is less than $2^d$. Our result significantly improves upon a recent construction, due to Dmitriy Zakharov, with size of order ${\phi }^d$ where $\phi =(1+\sqrt{5})/2\approx 1.618$ is the golden ratio.