Eppstein's bound on intersecting triangles revisited

TL;DR

This paper revisits Eppstein's bound on the number of triangles intersecting a point in a planar point set, providing a corrected proof of the original claim that relates the number of triangles and points.

Contribution

The paper offers a corrected proof of Eppstein's bound on intersecting triangles, fixing issues in the original proof and refining the theoretical understanding.

Findings

Confirmed the bound Omega(m^3 / (n^6 log^2 n)) for intersecting triangles

Provided a rigorous proof correcting Eppstein's original argument

Enhanced the theoretical framework for triangle-point intersection bounds

Abstract

Let S be a set of n points in the plane, and let T be a set of m triangles with vertices in S. Then there exists a point in the plane contained in Omega(m^3 / (n^6 log^2 n)) triangles of T. Eppstein (1993) gave a proof of this claim, but there is a problem with his proof. Here we provide a correct proof by slightly modifying Eppstein's argument.