Departmental Colloquium
Cosmin Pohoata
Emory University
The Heilbronn triangle problem
Abstract: Given an integer $n \geq 3$, the Heilbronn triangle problem asks for the smallest number $\Delta = \Delta(n)$ such that in every configuration of $n$ points in the unit square $[0,1]^2$ one can always find three among them which form a triangle of area at most $\Delta$. A trivial upper bound of the form $\Delta = O(1/n)$ follows from the simple observation that if one partitions $[0,1]^2$ into $n/2$ vertical strips of width $2/n$, then one of these strips must contain at least $3$ of the points. The problem of improving upon this basic observation is a difficult one, with a long and rich history. We will survey some of this history and discuss with some new connections between this problem and various themes in incidence geometry, harmonic analysis, and projection theory. Based on joint work with Alex Cohen and Dmitrii Zakharov.
Friday March 29, 2024 at 3:00 PM in 636 SEO