The right acute angles problem?

TL;DR

This paper explores a variant of the Danzer–Grünbaum acute angles problem, analyzing the asymptotic behavior of the maximum size of point sets with no large angles in high dimensions, and addresses related stability questions.

Contribution

It introduces a new limit-based variant of the problem, provides bounds for the asymptotic growth rate, and resolves a related stability problem while refuting a previous conjecture.

Findings

Established that the limit c is at least √2.

Provided bounds for the growth rate function F(α).

Solved a stability problem and refuted a conjecture.

Abstract

The Danzer--Gr\"unbaum 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 Gerencs\'er 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 Gerencs\'er and Harangi, the best lower bound we could get is $c\ge \sqrt 2$. We also solve a related problem of Erdos and F\"uredi on the "stability" of the acute angles problem and refute another conjecture stated in the same paper.