Richard P. Stanley Seminar in Combinatorics: Improved bounds for Heilbronn’s triangle problem and connections to projection theory

SEMINARS, HARVARD-MIT COMBINATORICS

Speaker:

Alex Cohen and Dmitrii Zakharov - MIT

Heilbronn’s triangle problem asks, how small is the smallest triangle formed by a set of points? Suppose n points are placed in the unit square and Delta is the smallest area triangle formed by three of these points. It is not too hard to see that Delta < C n^{-1}. Komlos, Pintz, and Szemeredi proved in 1981 that Delta << C n^{-8/7} by building on an ingenious method of Roth. We improve the bound to Delta < C n^{-8/7-ep}. The improvement comes from establishing new connections between Heilbronn's problem and projection theory.

All joint with Cosmin Pohoata.

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For more info, see https://math.mit.edu/combin/