View source for Heilbronn triangle problem

{{good article}}
{{Short description|On point sets with no small-area triangles}}
{{unsolved|mathematics|What is the asymptotic growth rate of the area of the smallest triangle determined by three out of <math>n</math> points in a square, when the points are chosen to maximize this area?}}
[[File:Heilbronn square n=6.svg|thumb|300px|Six points in the unit square, with the smallest triangles (red) having area 1/8, the optimal area for this number of points. Other larger triangles are colored blue. These points are an [[affine transformation]] of a [[regular hexagon]], but for larger numbers of points the optimal solution does not form a convex polygon.]]
In [[discrete geometry]] and [[discrepancy theory]], the '''Heilbronn triangle problem''' is a problem of placing points in the plane, avoiding [[triangle]]s of small [[area]]. It is named after [[Hans Heilbronn]], who [[conjecture]]d that, no matter how points are placed in a given area, the smallest triangle area will be at most [[Proportionality (mathematics)#Inverse proportionality|inversely proportional]] to the [[Square (algebra)|square]] of the number of points. His conjecture was proven false, but the [[Asymptotic analysis|asymptotic growth]] rate of the minimum triangle area remains unknown.

==Definition==
The Heilbronn triangle problem concerns the placement of <math>n</math> points within a shape in the plane, such as the [[unit square]] or the [[unit disk]], for a given {{nowrap|number <math>n</math>.}} Each triple of points form the three vertices of a [[triangle]], and among these triangles, the problem concerns the smallest triangle, as measured by area. Different placements of points will have different smallest triangles, and the problem asks: how should <math>n</math> points be placed to maximize the area of the smallest {{nowrap|triangle?{{r|roth}}}}

More formally, the shape may be assumed to be a [[compact set]] <math>D</math> in the plane, meaning that it stays within a bounded distance from the origin and that points are allowed to be placed on its boundary. In most work on this problem, <math>D</math> is additionally a [[convex set]] of nonzero area. When three of the placed points [[collinear|lie on a line]], they are considered as forming a [[degeneracy (mathematics)|degenerate]] triangle whose area is defined to be zero, so placements that maximize the smallest triangle will not have collinear triples of points. The assumption that the shape is compact implies that there exists an optimal placement of <math>n</math> points, rather than only a sequence of placements approaching optimality. The number <math>\Delta_D(n)</math> may be defined as the area of the smallest triangle in this optimal {{nowrap|placement.{{r|roth}}{{efn|Roth's definition uses slightly different notation, and [[Normalization (statistics)|normalizes]] the area of the triangle by dividing it by the area of <math>D</math>.}}}} An example is shown in the figure, with six points in a unit square. These six points form <math>\tbinom63=20</math> different triangles, four of which are shaded in the figure. Six of these 20 triangles, with two of the shaded shapes, have area 1/8; the remaining 14 triangles have larger areas. This is the optimal placement of six points in a unit square: all other placements form at least one triangle with area 1/8 or smaller. Therefore, {{nowrap|<math>\Delta_D(6)=\tfrac18</math>.{{r|goldberg}}}}

Although researchers have studied the value of <math>\Delta_D(n)</math> for specific shapes and specific small numbers of points,{{r|goldberg|comyeb|zenche}} Heilbronn was concerned instead about its [[asymptotic behavior]]: if the shape <math>D</math> is held fixed, but <math>n</math> varies, how does the area of the smallest triangle vary {{nowrap|with <math>n</math>?}} That is, Heilbronn's question concerns the growth rate {{nowrap|of <math>\Delta_D(n)</math>,}} as a function {{nowrap|of <math>n</math>.}} For any two shapes <math>D</math> {{nowrap|and <math>D'</math>,}} the numbers <math>\Delta_D(n)</math> and <math>\Delta_{D'}(n)</math> differ only by a constant factor, as any placement of <math>n</math> points within <math>D</math> can be scaled by an [[affine transformation]] to fit {{nowrap|within <math>D'</math>,}} changing the minimum triangle area only by a constant. Therefore, in bounds on the growth rate of <math>\Delta_D(n)</math> that omit the [[constant of proportionality]] of that growth, the choice of <math>D</math> is irrelevant and the subscript may be {{nowrap|omitted.{{r|roth}}}}

==Heilbronn's conjecture and its disproof==
Heilbronn conjectured prior to 1951 that the minimum triangle area always shrinks rapidly as a function {{nowrap|of <math>n</math>}}—more specifically, inversely proportional to the square {{nowrap|of <math>n</math>.{{r|roth}}{{efn|The conjecture is credited to Heilbronn in {{harvtxt|Roth|1951}}, but without citation to any specific publication.}}}} In terms of [[big O notation]], this can be expressed as the bound
<math display=block>\Delta(n)=O\left(\frac{1}{n^2}\right).</math>
[[File:No-three-in-line.svg|thumb|upright=0.8|Solutions to the [[no-three-in-line problem]], large sets of grid points with no three collinear points, can be scaled into a unit square with minimum triangle area {{nowrap|<math>\Omega(1/n^2)</math>.}}]]
In the other direction, [[Paul Erdős]] found examples of point sets with minimum triangle area proportional {{nowrap|to <math>1/n^2</math>,}} demonstrating that, if true, Heilbronn's conjectured bound could not be strengthened. Erdős formulated the [[no-three-in-line problem]], on large sets of grid points with no three in a line, to describe these examples. As Erdős observed, when <math>n</math> is a [[prime number]], the set of <math>n</math> points <math>(i,i^2\bmod n)</math> on an <math>n\times n</math> [[integer lattice|integer grid]] (for {{nowrap|<math>0\le i<n</math>)}} have no three collinear points, and therefore by [[Pick's formula]] each of the triangles they form has area at {{nowrap|least <math>\tfrac12</math>.}} When these grid points are scaled to fit within a unit square, their smallest triangle area is proportional {{nowrap|to <math>1/n^2</math>,}} matching Heilbronn's conjectured upper bound. If <math>n</math> is not prime, then a similar construction using a prime number close to <math>n</math> achieves the same asymptotic lower {{nowrap|bound.{{r|roth}}{{efn|Erdős's construction was published in {{harvtxt|Roth|1951}}, credited to Erdős.}}}}

{{harvtxt|Komlós|Pintz|Szemerédi|1982}} eventually disproved Heilbronn's conjecture, by using the [[probabilistic method]] to find sets of points whose smallest triangle area is larger than the ones found by Erdős. Their construction involves the following steps:
*Randomly place <math>n^{1+\varepsilon}</math> points in the unit square, for {{nowrap|some <math>\varepsilon>0</math>.}}
*Remove all pairs of points that are unexpectedly close together.
*Prove that there are few remaining low-area triangles and therefore only a sublinear number of cycles formed by two, three, or four low-area triangles. Remove all points belonging to these cycles.
*Apply a [[triangle removal lemma]] for 3-uniform [[hypergraph]]s of high [[girth (graph theory)|girth]] to show that, with high probability, the remaining points include a subset of <math>n</math> points that do not form any small-area triangles.
The area resulting from their construction grows asymptotically as{{r|kps82}}
<math display=block>\Delta(n)=\Omega\left(\frac{\log n}{n^2}\right).</math>
The proof can be [[Derandomization|derandomized]], leading to a [[polynomial time|polynomial-time]] algorithm for constructing placements with this triangle area.{{r|bkhl}}

==Upper bounds==
Every set of <math>n</math> points in the unit square forms a triangle of area at most inversely proportional {{nowrap|to <math>n</math>.}} One way to see this is to [[Point set triangulation|triangulate]] the [[convex hull]] of the given point {{nowrap|set <math>S</math>,}} and choose the smallest of the triangles in the triangulation. Another is to sort the points by their {{nowrap|<math>x</math>-coordinates,}} and to choose the three consecutive points in this ordering whose {{nowrap|<math>x</math>-coordinates}} are the closest together. In the first paper published on the Heilbronn triangle problem, in 1951, [[Klaus Roth]] proved a stronger upper bound {{nowrap|on <math>\Delta(n)</math>,}} of the form{{r|roth}}
<math display=block>\Delta(n)=O\left(\frac{1}{n\sqrt{\log\log n}}\right).</math>
The best bound known to date is of the form
<math display=block>\Delta(n)\leq\frac{\exp{\left(c\sqrt{\log n}\right)}}{n^{8/7}},</math>
for some {{nowrap|constant <math>c</math>,}} proven by {{harvtxt|Komlós|Pintz|Szemerédi|1981}}.{{r|kps81}}

A new upper bound equal to <math>n^{-\frac{8}{7}-\frac{1}{2000}}</math> was proven by {{harvtxt|Cohen|Pohoata|Zakharov|2023}}.{{r|cpz23|sloman}}

==Specific shapes and numbers==
{{harvtxt|Goldberg|1972}} has investigated the optimal arrangements of <math>n</math> points in a square, for <math>n</math> up to 16.{{r|goldberg}} Goldberg's constructions for up to six points lie on the boundary of the square, and are placed to form an [[affine transformation]] of the vertices of a [[regular polygon]]. For larger values {{nowrap|of <math>n</math>,}} {{harvtxt|Comellas|Yebra|2002}} improved Goldberg's bounds, and for these values the solutions include points interior to the square.{{r|comyeb}} These constructions have been proven optimal for up to seven points. The proof used a computer search to subdivide the [[Configuration space (mathematics)|configuration space]] of possible arrangements of the points into 226 different subproblems, and used [[nonlinear programming]] techniques to show that in 225 of those cases, the best arrangement was not as good as the known bound. In the remaining case, including the eventual optimal solution, its optimality was proven using [[symbolic computation]] techniques.{{r|zenche}}

The following are the best known solutions for 7–12 points in a unit square, found through [[simulated annealing]];{{r|comyeb}} the arrangement for seven points is known to be optimal.{{r|zenche}}

<gallery>
Heilbronn triangles, 7 points in square.svg|7 points in a square, all 8 minimal triangles shaded {{nowrap|(<math>A\approx 0.0839</math>)}}
Heilbronn triangles, 8 points in square.svg|8 points in a square, 5 of 12 minimal triangles shaded{{efn|name=mofn|Where several minimal-area triangles can be shown without calculation to be equal in area, only one of them is shaded.}} {{nowrap|(<math>A\approx 0.0724</math>)}}
Heilbronn triangles, 9 points in square.svg|9 points in a square, 6 of 11 minimal triangles shaded{{efn|name=mofn}} {{nowrap|(<math>A\approx 0.0549</math>)}}
Heilbronn triangles, 10 points in square.svg|10 points in a square, 3 of 16 minimal triangles shaded{{efn|name=mofn}} {{nowrap|(<math>A\approx 0.0465</math>)}}
Heilbronn triangles, 11 points in square.svg|11 points in a square, 8 of 28 minimal triangles shaded{{efn|name=mofn}} {{nowrap|(<math>A\approx 0.0370</math>)}}
Heilbronn triangles, 12 points in square.svg|12 points in a square, 3 of 20 minimal triangles shaded{{efn|name=mofn}} {{nowrap|(<math>A\approx 0.0326</math>)}}
</gallery>

Instead of looking for optimal placements for a given shape, one may look for an optimal shape for a given number of points. Among convex shapes <math>D</math> with area one, the [[regular hexagon]] is the one that {{nowrap|maximizes <math>\Delta_D(6)</math>;}} for this shape, {{nowrap|<math>\Delta_D(6)=\tfrac16</math>,}} with six points optimally placed at the hexagon vertices.{{r|dyz}} The convex shapes of unit area that maximize <math>\Delta_D(7)</math> have {{nowrap|<math>\Delta_D(7)=\tfrac19</math>.{{r|yanzen}}}}

==Variations==
There have been many variations of this problem 
including the case of a uniformly random set of points, for which arguments based on either [[Kolmogorov complexity]] or [[Poisson approximation]] show that the [[expected value]] of the minimum area is inversely proportional to the cube of the number of points.{{r|jlv|grimmett}} Variations involving the volume of higher-dimensional [[simplex|simplices]] have also been studied.{{r|brass|lefmann|barnao}}

Rather than considering simplices, another higher-dimensional version adds another {{nowrap|parameter <math>k</math>,}} and asks for placements of <math>n</math> points in the unit [[hypercube]] that maximize the minimum volume of the [[convex hull]] of any subset of <math>k</math> points. For <math>k=d+1</math> these subsets form simplices but for larger values {{nowrap|of <math>k</math>,}} relative {{nowrap|to <math>d</math>,}} they can form more complicated shapes. When <math>k</math> is sufficiently large relative {{nowrap|to <math>\log n</math>,}} randomly placed point sets have minimum {{nowrap|<math>k</math>-point}} convex hull {{nowrap|volume <math>\Omega(k/n)</math>.}} No better bound is possible; any placement has <math>k</math> points with {{nowrap|volume <math>O(k/n)</math>,}} obtained by choosing some <math>k</math> consecutive points in coordinate order. This result has applications in [[range searching]] data structures.{{r|chazelle}}

==See also==
*[[Danzer set]], a set of points that avoids empty triangles of large area

==Notes==
{{notelist}}

==References==
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==External links==
*{{mathworld|id=HeilbronnTriangleProblem|title=Heilbronn Triangle Problem|mode=cs2}}
*[https://erich-friedman.github.io/packing/index.html Erich's Packing Center], by Erich Friedman, including the best known solutions to the Heilbronn problem for small values of <math>n</math> for squares, circles, equilateral triangles, and convex regions of variable shape but fixed area

[[Category:Discrete geometry]]
[[Category:Triangle problems]]
[[Category:Area]]
[[Category:Discrepancy theory]]