Extremal problems in combinatorial geometry
Handbook of combinatorics (vol. 1)
Crossing Numbers and Hard Erdös Problems in Discrete Geometry
Combinatorics, Probability and Computing
On Distinct Distances from a Vertex of a Convex Polygon
Discrete & Computational Geometry
On the number of tetrahedra with minimum, unit, and distinct volumes in three-space
SODA '07 Proceedings of the eighteenth annual ACM-SIAM symposium on Discrete algorithms
Proceedings of the nineteenth annual ACM-SIAM symposium on Discrete algorithms
On the number of tetrahedra with minimum, unit, and distinct volumes in three-space
Combinatorics, Probability and Computing
Extremal problems on triangle areas in two and three dimensions
Proceedings of the twenty-fourth annual symposium on Computational geometry
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Erdős, Purdy, and Straus conjectured that the number of distinct (nonzero) areas of the triangles determined by nnoncollinear points in the plane is at least $\lfloor \frac{n-1}{2} \rfloor$, which is attained for 茂戮驴n/ 2茂戮驴 and respectively $\lfloor n/2\rfloor$ equally spaced points lying on two parallel lines. We show that this number is at least $\frac{17}{38}n -O(1) \approx 0.4473n$. The best previous bound, $(\sqrt{2}-1)n-O(1)\approx 0.4142n$, which dates back to 1982, follows from the combination of a result of Burton and Purdy [5] and Ungar's theorem [23] on the number of distinct directions determined by nnoncollinear points in the plane.