2019/10/02 by Kupavskii, Andrey, Zakharov, Dmitriy · 1 citation
#Combinatorics (math.CO) #FOS: Mathematics #Metric Geometry (math.MG)
paper · doi:10.48550/arxiv.1910.00798
The Danzer--Grünbaum acute angles problem asks for the largest size of a set of points in \mathbb Rd 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 2d-1+1 and 2d-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(α) = limd→ ∞ f(d,α)1/d, where f(d,α) is the largest set of points in \mathbb Rd with no angle greater than α. Then the question is to find c:= limα→ π/2-F(α). 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≥ √ 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.