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 $$\varphi ^d$$ where $$\varphi = (1+\sqrt{5})/2 \approx 1.618$$ is the golden ratio.