Heilbronn Triangle Problem

• The Heilbronn triangle problem is a classic combinatorial geometry topic that seeks the maximum minimal triangle area formed by n points in a convex unit area.
• Recent advances have refined asymptotic bounds via analytic, combinatorial and computational approaches, incorporating Fourier analysis and incidence geometry.
• Methodological innovations, including optimization techniques and computational certification for small n, have provided sharper estimates and improved understanding of geometric configurations.

The Heilbronn triangle problem concerns the extremal behavior of the smallest possible area of a triangle determined by $n$ points constrained to a convex region of the Euclidean plane, usually a unit-area square, triangle, or disk. It asks: for a given $n$, how large can one guarantee the minimal area of any triangle formed by three of these points, as the arrangement ranges over all possible $n$-point configurations in the region? This classical problem lies at the intersection of combinatorial geometry, extremal analysis, and computational optimization, and has driven a series of methodological and theoretical innovations for over seventy years.

1. Origins, Classical Formulation, and Notation

Proposed by H. A. Heilbronn circa 1950, the original conjecture postulated that, for any configuration of $n$ points in a convex planar domain of unit area (typically the unit square $[0,1]^2$), any placement will yield a triangle of area at most $O(n^{-2})$; that is, $h(n) = O(n^{-2})$ is the asymptotic supremum of the minimum triangle area over all $n$-point sets with no three points collinear. Explicitly, let

$$h(n) = \max_{P \subset [0,1]^2,\, |P|=n}\, \min_{\substack{x, y, z \in P \ \text{not collinear}}} \mathrm{Area}(\triangle xyz).$$

This "max–min" structure—maximize the minimum area across all valid configurations—has analogous formulations in the unit disk, in the unit triangle, and in higher dimensions for simplices.

2. Historical Progression of Bounds

Asymptotic and Small-$n$ Results

Early work leveraged pigeonhole and packing arguments to show the trivial upper bound $h(n) = O(n^{-2})$.

• Roth (1951, 1972) introduced analytic techniques, establishing $h(n) = O(n^{-1}(\log \log n)^{-1/2})$ (Cohen et al., 2023).
• Schmidt (1972) improved this to $h(n) = O(n^{-1}(\log n)^{-1/2})$.
• Komlós, Pintz, and Szemerédi (1982) established the first polynomial refinement, $h(n) \leq \exp(c\sqrt{\log n})\, n^{-8/7}$ for some absolute $c>0$.

For small $n$, exact values (or best-known conjectural values) are tabulated below (0911.4375, Monji et al., 16 Dec 2025):

$n$	$h(n)$ (best-known)	Method/Source
3	$0.5$	Trivial (right triangle)
4	$0.25$	Square corners/partition
5	$\approx 0.19245$	$\sqrt{3}/9$; Lu et al. 1991
6	$0.125$	$1/8$; Lu et al. 1991
7	$0.0838591$	Zeng & Chen 2008
8	$0.0723764$	Dehbi & Zeng 2022
9	$0.0548767$	(Monji et al., 16 Dec 2025)

Recent Upper and Lower Bounds

The upper bound has seen recent improvements:

• Cohen, Pohoata, and Zakharov (2023) proved, for sufficiently large $n$, that

$h(n) \leq C_{2}\, n^{-2} (\ln \ln n)^2$

and for all large $n$,

$h(n) \leq n^{-8/7-1/2000}$

improving classical exponents (Cohen et al., 2023, Zakharov, 2022).

Lower bounds remain less sharp, but significant progress exists:

• Komlós–Pintz–Szemerédi: $h(n) \geq \Omega(n^{-2} \log n)$ (Ellmann, 2017).
• Ellmann (2024): constructed sets with all triangle areas at least $\Omega(n^{-3/2} (\log n)^{-7/2})$, representing a polynomial improvement over earlier constructions (Ellmann, 2017).
• Agama (2024): further inserted a logarithmic factor, giving $h(n) \gg (\log n)/n^{3/2}$ (Agama, 2020).

3. Methodological Advances: Analytic, Combinatorial, and Algorithmic Techniques

Fourier-Analytic and Density-Increment Methods

• Roth's analytic approach and subsequent refinements by Schmidt built on the density-increment method and Fourier analysis, establishing early nontrivial upper bounds (Cohen et al., 2023).
• Modern approaches, as in (Cohen et al., 2023), combine projection-theoretic arguments (discretized Marstrand/Orponen–Shmerkin–Wang theorems) with incidence geometry and hypergraph removal to avoid logarithmic losses and to incrementally improve exponents.

Incidence Geometry

• The “high–low method” of Guth–Solomon–Wang facilitates multiscale propagation of incidence concentrations, enabling conversion of local regularity (Frostman-type conditions) into global small-area guarantees (Cohen et al., 2023).
• Key technical ingredients include normalized incidence counts, local packing numbers, and smoothed counts over lines and strips.

Optimization and Computational Approaches

For small $n$, verified optima are obtained via global optimization frameworks:

• Formulation as a mixed-integer quadratically constrained programming (MIQCP) or nonconvex QCP (Monji et al., 16 Dec 2025).
• Structural enhancements: bound tightening, symmetry breaking, boundary occupancy constraints, and local packing inequalities drastically prune the feasible set and speed up computation.
• The approach of (Monji et al., 16 Dec 2025) certified optima up to $n=9$, solving the $n=9$ instance (to proven optimality) in $\approx 10$ minutes, compared to a prior $31$-day GPGPU grid search.
• Exhaustive grid-based, integer-arithmetic proofs verify upper bounds for $N=5,6,7$, with the possibility of extension via high-performance computing (0911.4375).

New Geometric Constructions

• Ellmann's lower-bound configuration uses prime-indexed regular polygons inscribed on concentric circles, with exclusion arcs to avoid small triangles, then projects to the unit circle, yielding improved (logarithmically penalized) lower bounds (Ellmann, 2017).
• Agama’s “geometry of compression” utilizes anisotropic rescaling (compression maps) and precise covering arguments to close the log-factor gap (Agama, 2020).

4. Higher-Dimensional and Topological Variants

Generalization to Higher Dimensions

• Zakharov (Zakharov, 2022) developed a recursive bound for the minimal volume of simplices generated by $n$ points in $[0,1]^d$ ($d\geq 3$). The orthogonal decomposition lemma yields, e.g., $A_{4,3}(n) = O(n^{-3})$ in dimension 3, with a recursive bound based on projection and partitioning.
• For the planar case $(d=2)$, the method does not yield an improvement over the best analytic result, but the recursion has substantial impact for $d\geq 3$.

Topological and Combinatorial Extensions

• A $\mathbb{Z}_2$-variant replaces geometric triangles by cycles in topological graphs: for any drawing of $K_n$, all nonzero $\mathbb{Z}_2$–cycles can be forced to have area at least $O(1)$, showing a strict dichotomy with the Euclidean problem where minimal areas decay rapidly as $n$ increases (Hubard et al., 2022).
• The distinction between geometric and topological behavior signals crucial limitations on homological methods for approaching the original extremal problem, especially regarding minimum triangle area asymptotics.

5. Current Frontiers: Open Problems, Limitations, and Comparisons

Major gaps remain between upper and lower asymptotic bounds. The best known exponents differ by powers of $n$, and logarithmic factors remain difficult to eliminate. Key unresolved problems include:

• Closing the exponent gap between $O(n^{-2})$ and the best-constructed lower bounds ($\gg n^{-3/2}(\log n)^{-7/2}$ or $h(n) = \Omega((\log n)/n^{3/2})$) (Ellmann, 2017, Agama, 2020).
• Determining whether configurations maximizing $h(n)$ must have boundary points, as suggested by computational evidence (0911.4375, Monji et al., 16 Dec 2025).
• Extending rigorous computational certification beyond $n=9$ in both square and triangle cases (Monji et al., 16 Dec 2025).
• Potential enhancements of analytic and projection-theoretic methods to remove remaining logarithmic losses and to establish $O(n^{-2})$ upper bounds without extra factors (Zakharov, 2022, Cohen et al., 2023).
• Adapting homological, parity-based, or combinatorial-topological methods to yield new geometric insight or tighter bounds for the strictly Euclidean problem (Hubard et al., 2022).

Comparisons with related extremal problems (e.g., Motzkin–Schmidt, smallest convex $k$-gon areas, strip covering) indicate that the philosophical and methodological tools developed for the triangle problem are broadly influential across incidence geometry and extremal combinatorics.

6. Computational Data and Explicit Bounds for Small $n$

Verified and conjectured optimal values for small $n$, as well as rigorous algorithmic upper bounds, are summarized below (0911.4375, Monji et al., 16 Dec 2025):

$n$	Certified $h(n)$	Reference or Method
3	$0.5$	Trivial right triangle
4	$0.25$	Square/triangle corners
5	$\sqrt{3}/9 \approx 0.19245$	Lu et al. 1991
6	$0.125$	Lu et al. 1991
7	$0.0838591$	Zeng & Chen 2008
8	$0.0723764$	Dehbi & Zeng 2022
9	$0.0548767$	(Monji et al., 16 Dec 2025), certified

For small $N \leq 7$, automated-combinatorial and grid-based methods yield rigorous upper bounds generally matching or improving prior estimates (0911.4375).

7. Outlook and Broader Impact

The Heilbronn triangle problem continues to generate new methods spanning pure, computational, and applied aspects of extremal geometry. Potential further advances may arise from:

• Fusion of incidence geometry with analytic and arithmetic tools.
• More scalable and automated computer-aided proof systems for certifying extremal configurations in higher-$n$ or higher-dimensional settings.
• Cross-pollination of methods with related combinatorial-geometric problems, such as sum-product phenomena, general small-volume simplex problems, and the analysis of geometric configurations in statistical physics and optimization.

The problem remains a central benchmark for methodologies in combinatorial geometry and a nexus for deeper connections between analysis, number theory, and computational geometry.

References:

(Ellmann, 2017, Cohen et al., 2023, Monji et al., 16 Dec 2025, Agama, 2020, Zakharov, 2022, 0911.4375, Hubard et al., 2022)