An Improved Bound on the Number of Unit Area Triangles

TL;DR

This paper improves the upper bound on the maximum number of triangles with unit area that can be formed by n points in the plane, refining previous bounds with a tighter asymptotic estimate.

Contribution

It presents a new bound of O(n^{9/4+ε}) on the number of unit-area triangles, advancing the understanding of geometric combinatorics.

Findings

New upper bound of O(n^{9/4+ε}) for unit-area triangles

Improves previous bound of O(n^{44/19})

Advances theoretical understanding of geometric configurations

Abstract

We show that the number of unit-area triangles determined by a set of $n$ points in the plane is $O(n^{9/4+\epsilon})$, for any $\epsilon>0$, improving the recent bound $O(n^{44/19})$ of Dumitrescu et al.