The right acute angles problem?

Abstract

The Danzer--Grünbaum acute angles problem asks for the largest size of a set of points in ${\mathbb{R}}^d$ that determines only acute angles. Recently, the problem was essentially solved thanks to the results of the second author and of Gerencsér and Harangi: now, the lower and the upper bounds are $2^{d-1}+1$ and $2^d-1$, respectively. The lower-bound construction is surprisingly simple. In this note, we suggest the following variant of the problem, which is one way to "save" the problem. Put $F(\alpha )={\lim }_{d\to \infty }f(d,\alpha {)}^{1/d}$, where $f(d,\alpha )$ is the largest set of points in ${\mathbb{R}}^d$ with no angle greater than $\alpha$. Then the question is to find $c:={\lim }_{\alpha \to \pi /2^{-}}F(\alpha ).$ Although one may expect that $c=2$ in view of the result of Gerencsér and Harangi, the best lower bound we could get is $c\ge \sqrt{2}$. We also solve a related problem of Erdos and Füredi on the "stability" of the acute angles problem and refute another conjecture stated in the same paper.

Minor corrections