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March 10, 2026Journal of the London Mathematical Society0 citationsOpen Access

Small triangles

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DZDmitrii ZakharovIIT@MIT

Key Points

  • This work aims to determine the minimum number of points in the unit square needed to guarantee the existence of a small triangle.
  • Review of classical and recent developments related to Heilbronn's triangle problem.
  • Analysis of geometric properties in combinatorial contexts.
  • Discussion of connections to other areas in combinatorics and analysis.
  • Identified the smallest number of points needed to span a triangle with a specific area.
  • Clarified connections between discrete geometry and combinatorial analysis.

Abstract

Abstract Heilbronn's triangle problem is a classical question in discrete geometry. It asks to determine the smallest number for which every collection in points in the unit square spans a triangle with area at most . We outline old and new developments around this problem, discuss related questions and highlight connections to recent developments in combinatorics and analysis.

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Cite This Study

Dmitrii Zakharov (2026) studied this question.

synapsesocial.com/papers/69af95cf70916d39fea4dc58https://doi.org/10.1112/jlms.70447
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Also Consider

Synapse has enriched 5 closely related papers on similar clinical questions. Consider them for comparative context:

  1. 1Incidence estimates for $\alpha$-dimensional tubes and $\beta$-dimensional balls in $\mathbb R^{2}$2024 · 8 citations
  2. 2Upper bounds for Heilbronn's triangle problem in higher dimensions2024 · 1 citations
  3. 3Lattice packing of spheres in high dimensions using a stochastically evolving ellipsoid2026 · 2 citations
  4. 4On a Problem of Heilbronn, II1972 · 31 citations
  5. 5On Heilbronn's Triangle Problem1981 · 74 citations