A power improvement in the Heilbronn triangle lower bound

1 · The essentials

Place a large number of points inside a square of side length one, and look at every triangle made by choosing three of them. The challenge is to arrange the points so that even the smallest triangle is as large as possible. The main result says that, for every sufficiently large number of points, there is an arrangement in which every triangle has area at least $c_{1} n^{- 2 + η}$ . Here $n$ is the number of points, and $c_{1}$ and $η$ are fixed positive numbers that do not change with $n$ . Although $η$ is extremely small, its positivity rules out the proposed almost- $n^{- 2}$ upper bound described in the source. The main result has a Lean formalization, though the supplied material does not specify its detailed scope.

A picture to help (an imperfect image, not to be taken literally)

As an imperfect image, imagine placing dots on a square sheet of paper and having someone inspect every possible three-dot triangle, looking for the smallest one. You want to place the dots so that even their best attempt finds a triangle above your guaranteed area; the result gives a stronger guarantee when there are sufficiently many dots.

Why it matters

The improvement changes the power of $n$ , rather than merely multiplying the same bound by a larger fixed number. According to the source, this is enough to refute the almost- $n^{- 2}$ upper-bound formulation of Heilbronn's triangle problem. It does not give a complete answer to how large the smallest triangle can be.

2 · Going further

Heilbronn's triangle problem asks how large one can make the smallest triangle determined by $n$ points in the unit square. The result states that there exist absolute constants $η, c_{1} > 0$ such that, for every sufficiently large integer $n$ , one can choose $n$ points for which every determined triangle has area at least $c_{1} n^{- 2 + η}$ . The guarantee applies to every choice of three points, not just to selected triangles.

The context supplied is a conjectured almost- $n^{- 2}$ upper bound. The manuscript states that its lower bound makes that formulation false. Compared with $n^{- 2}$ , the new lower bound has the factor $c_{1} n^{η}$ : the exponent improvement is fixed and positive, although the abstract emphasizes that $η$ is extremely small. The source does not give a precise quantified definition of the phrase “almost- $n^{- 2}$ .”

The repository summary describes a construction of point sets, but the supplied abstract gives no construction details or argument outline. Consequently, the method behind the improvement cannot be explained from this material alone. Only the main manuscript is supplied, so no relationship to companion manuscripts can be established here. The main result has a Lean formalization; the available material does not identify the exact scope of that verification.

Useful notions

Area of a triangle
The unit square
Powers and exponents
Lower bounds and upper bounds
Constants independent of the number of points

What was known before / what is new

Before

The source describes Heilbronn's triangle problem as asking how large the smallest determined triangle can be. It identifies an almost- $n^{- 2}$ upper bound as a conjectured formulation. No earlier quantitative lower bounds or historical details are supplied.

New

The main result provides point sets whose every triangle has area at least $c_{1} n^{- 2 + η}$ , for fixed positive absolute constants and all sufficiently large $n$ . The improvement is in the exponent, with $η$ extremely small but positive. The manuscript states that this refutes the conjectured almost- $n^{- 2}$ upper-bound formulation.

Limits and assumptions

The statement concerns the unit square and sufficiently large integers $n$ , not every possible number of points. It guarantees the existence of suitable arrangements, not the bound for arbitrary arrangements. Neither numerical values for the constants nor a threshold for $n$ are supplied. The main result has a Lean formalization, but its detailed scope is not specified, so verification should not be extended to every surrounding claim. The supplied material does not give an optimal bound or a full resolution of how large the smallest triangle can be.

Mini glossary

Unit square: A square with side length one.
Determined triangle: A triangle formed by choosing three points from the given arrangement.
Lower bound: A guaranteed minimum for a quantity, here the area of every triangle in a suitably chosen arrangement.
Absolute constant: A fixed number that does not depend on the number of points $n$ .
Exponent improvement: The replacement of the power $n^{- 2}$ by $n^{- 2 + η}$ with a fixed positive $η$ in the lower bound.

Keywords : Heilbronn triangle problem · Geometric combinatorics · Exponent improvement

AI-generated explanation based on the manuscript. Unofficial. Refer to the manuscript for the authoritative version.

3 · Technical details and sources

Claim made in the manuscript (source text)

For every sufficiently large n, constructs n points in the unit square such that every triangle has area at least $n^{- 2 + c}$ for one absolute c > 0. This disproves the conjectured almost-n⁻² upper bound in Heilbronn's triangle problem, which asks how large the smallest determined triangle can be.

Verified by Lean: the Lean software checked the part of the reasoning described below, sometimes only part of the manuscript. This does not mean independent mathematicians have validated the result.

Scope of the Lean formalization (source text)

…

lean/docs/191.md

Manuscripts in this family

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A power improvement in the Heilbronn triangle lower bound

25/09/2026

To be verified

There are absolute constants $η, c_{1} > 0$ such that, for every sufficiently large integer n, one can choose n points in the unit square so that every triangle they determine has area at least $c_{1} n^{- 2 + η}$ . Thus the almost n⁻² upper-bound formulation of Heilbronn's triangle problem is false. The exponent η is fixed but extremely small.

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