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
Extremal problems on triangle areas in two and three dimensions
Journal of Combinatorial Theory Series A
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We prove a conjecture of Erdős, Purdy, and Straus on the number of distinct areas of triangles determined by a set of $n$ points in the plane. We show that if $P$ is a set of $n$ points in the plane, not all on one line, then $P$ determines at least $\lfloor\frac{n-1}{2}\rfloor$ triangles with pairwise distinct areas. Moreover, one can find such $\lfloor\frac{n-1}{2}\rfloor$ triangles all sharing a common edge.