2025/10/30 by Maldague, Dominique, Wang, Hong, Zakharov, Dmitrii
#Classical Analysis and ODEs (math.CA) #Combinatorics (math.CO) #FOS: Mathematics #Metric Geometry (math.MG)
paper · doi:10.48550/arxiv.2510.26644
We show that among any n points in the unit cube one can find a triangle of area at most n-2/3-c for some absolute constant c >0. This gives the first non-trivial upper bound for the three-dimensional version of Heilbronn's triangle problem. This estimate is a consequence of the following result about configurations of point-line pairs in \mathbb R3: for n ≥ 2 let p1, …,pn ∈ [0,1]3 be a collection of points and let ℓi be a line through pi for every i such that d(pi, ℓj) ≥ δ for all i≠ j. Then we have n \lesssim δ-3+γ for some absolute constant γ>0. The analogous result about point-line configurations in the plane was previously established by Cohen, Pohoata and the last author.