A new upper bound for the Heilbronn triangle problem was proved by Alex Cohen, Cosmin Pohoata, and Dmitrii Zakharov. Congratulations!
The paper is
A new upper bound for the Heilbronn triangle problem
Abstract: For sufficiently large n, we show that in every configuration of n points chosen inside the unit square there exists a triangle of area less than Image may be NSFW.
Clik here to view.This improves upon a result of Komlós, Pintz and Szemerédi from 1982. Our approach establishes new connections between the Heilbronn triangle problem and various themes in incidence geometry and projection theory which are closely related to the discretized sum-product phenomenon.
The introduction of the paper beautifully presents the rich history of this problem and some developments and connections leading to the new results. Heilbronn asked in the late 1940’s what is the smallest area of a triangle formed by three point in a configuration of Image may be NSFW.
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points in the unit square, and Komlós, Pintz and Szemerédi’s (KPS) 1982 paper presented the state of our knowledge until a month or so ago. KPS disproved a conjecture of Heilbronn that there is always a triangle of area Image may be NSFW.
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. As for upper bounds they improved methods by Roth (who wrote a series of papers on the problem from 1950 to 1970), and showed that there is always a triangle of area Image may be NSFW.
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. Using a variety of recent techniques and connections Cohen, Pohoata, and Zakharov improved the exponent.
