Abstract
Erdős, Purdy, and Straus conjectured that the number of distinct (nonzero) areas of the triangles determined by n noncollinear points in the plane is at least \(\lfloor \frac{n-1}{2} \rfloor\), which is attained for ⌈n / 2⌉ 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 n noncollinear points in the plane.
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Dumitrescu, A., Tóth, C.D. (2007). Distinct Triangle Areas in a Planar Point Set. In: Fischetti, M., Williamson, D.P. (eds) Integer Programming and Combinatorial Optimization. IPCO 2007. Lecture Notes in Computer Science, vol 4513. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-72792-7_10
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DOI: https://doi.org/10.1007/978-3-540-72792-7_10
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