Heilbronn triangle problem

The Heilbronn triangle problem concerns the placement of $n$ points within a shape in the plane, such as the unit square or the unit disk, for a given {{nowrap|number $n$.}} 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 $n$ 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 $D$ 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, $D$ is additionally a convex set of nonzero area. When three of the placed points lie on a line, they are considered as forming a 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 $n$ points, rather than only a sequence of placements approaching optimality. The number $\backslash Delta\_D(n)$ 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 normalizes the area of the triangle by dividing it by the area of $D$.}}}} An example is shown in the figure, with six points in a unit square. These six points form $\backslash tbinom63=20$ 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|$\backslash Delta\_D(6)=\backslash tfrac18$.{{r|goldberg}}}}

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

Heilbronn conjectured prior to 1951 that the minimum triangle area always shrinks rapidly as a function {{nowrap|of $n$}}—more specifically, inversely proportional to the square {{nowrap|of $n$.{{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

$$\backslash Delta(n)=O\backslash left(\backslash frac\{1\}\{n^2\}\backslash right).$$

File:No-three-in-line.svg, large sets of grid points with no three collinear points, can be scaled into a unit square with minimum triangle area {{nowrap|$\backslash Omega(1/n^2)$.}}]]

In the other direction, Paul Erdős found examples of point sets with minimum triangle area proportional {{nowrap|to $1/n^2$,}} 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 $n$ is a prime number, the set of $n$ points $(i,i^2\backslash bmod\; n)$ on an $n\backslash times\; n$ integer grid (for {{nowrap|

{{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 $n^\{1+\backslash varepsilon\}$ points in the unit square, for {{nowrap|some $\backslash varepsilon>0$.}}
• 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 hypergraphs of high girth to show that, with high probability, the remaining points include a subset of $n$ points that do not form any small-area triangles.

The area resulting from their construction grows asymptotically as{{r|kps82}}

$$\backslash Delta(n)=\backslash Omega\backslash left(\backslash frac\{\backslash log\; n\}\{n^2\}\backslash right).$$

The proof can be derandomized, leading to a polynomial-time algorithm for constructing placements with this triangle area.{{r|bkhl}}

Every set of $n$ points in the unit square forms a triangle of area at most inversely proportional {{nowrap|to $n$.}} One way to see this is to triangulate the convex hull of the given point {{nowrap|set $S$,}} and choose the smallest of the triangles in the triangulation. Another is to sort the points by their {{nowrap|$x$-coordinates,}} and to choose the three consecutive points in this ordering whose {{nowrap|$x$-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 $\backslash Delta(n)$,}} of the form{{r|roth}}

$$\backslash Delta(n)=O\backslash left(\backslash frac\{1\}\{n\backslash sqrt\{\backslash log\backslash log\; n\}\}\backslash right).$$

The best bound known to date is of the form

$$\backslash Delta(n)\backslash leq\backslash frac\{\backslash exp\{\backslash left(c\backslash sqrt\{\backslash log\; n\}\backslash right)\}\}\{n^\{8/7\}\},$$

for some {{nowrap|constant $c$,}} proven by {{harvtxt|Komlós|Pintz|Szemerédi|1981}}.{{r|kps81}}

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

{{harvtxt|Goldberg|1972}} has investigated the optimal arrangements of $n$ points in a square, for $n$ 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 $n$,}} {{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 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}}

Heilbronn triangles, 7 points in square.svg|7 points in a square, all 8 minimal triangles shaded {{nowrap|($A\backslash approx\; 0.0839$)}}

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|($A\backslash approx\; 0.0724$)}}

Heilbronn triangles, 9 points in square.svg|9 points in a square, 6 of 11 minimal triangles shaded{{efn|name=mofn}} {{nowrap|($A\backslash approx\; 0.0549$)}}

Heilbronn triangles, 10 points in square.svg|10 points in a square, 3 of 16 minimal triangles shaded{{efn|name=mofn}} {{nowrap|($A\backslash approx\; 0.0465$)}}

Heilbronn triangles, 11 points in square.svg|11 points in a square, 8 of 28 minimal triangles shaded{{efn|name=mofn}} {{nowrap|($A\backslash approx\; 0.0370$)}}

Heilbronn triangles, 12 points in square.svg|12 points in a square, 3 of 20 minimal triangles shaded{{efn|name=mofn}} {{nowrap|($A\backslash approx\; 0.0326$)}}

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 $D$ with area one, the regular hexagon is the one that {{nowrap|maximizes $\backslash Delta\_D(6)$;}} for this shape, {{nowrap|$\backslash Delta\_D(6)=\backslash tfrac16$,}} with six points optimally placed at the hexagon vertices.{{r|dyz}} The convex shapes of unit area that maximize $\backslash Delta\_D(7)$ have {{nowrap|$\backslash Delta\_D(7)=\backslash tfrac19$.{{r|yanzen}}}}

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 simplices have also been studied.{{r|brass|lefmann|barnao}}

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

• Danzer set, a set of points that avoids empty triangles of large area

{{notelist}}

{{reflist|refs=

{{citation

| last1 = Barequet | first1 = Gill

| last2 = Naor | first2 = Jonathan

| issue = 3

| journal = Far East Journal of Applied Mathematics

| mr = 2283483

| pages = 343–354

| title = Large $k$-D simplices in the $d$-dimensional unit cube

| volume = 24

| year = 2006}}

{{citation

| last1 = Bertram-Kretzberg | first1 = Claudia

| last2 = Hofmeister | first2 = Thomas

| last3 = Lefmann | first3 = Hanno

| doi = 10.1137/S0097539798348870

| issue = 2

| journal = SIAM Journal on Computing

| mr = 1769363

| pages = 383–390

| title = An algorithm for Heilbronn's problem

| volume = 30

| year = 2000| hdl = 2003/5313

| hdl-access = free

}}

{{citation

| last = Brass | first = Peter

| doi = 10.1137/S0895480103435810

| issue = 1

| journal = SIAM Journal on Discrete Mathematics

| mr = 2178353

| pages = 192–195

| title = An upper bound for the $d$-dimensional analogue of Heilbronn's triangle problem

| volume = 19

| year = 2005}}

{{citation

| last = Chazelle | first = Bernard | author-link = Bernard Chazelle

| isbn = 978-0-521-00357-5

| page = 266

| publisher = Cambridge University Press

| title = The Discrepancy Method: Randomness and Complexity

| url = https://books.google.com/books?id=dmOPmEh6LdYC&pg=PA266

| year = 2001}}

{{citation

| last1 = Comellas | first1 = Francesc

| last2 = Yebra | first2 = J. Luis A.

| doi = 10.37236/1623 | doi-access = free

| issue = 1

| journal = Electronic Journal of Combinatorics

| mr = 1887087

| page = R6

| title = New lower bounds for Heilbronn numbers

| volume = 9

| year = 2002}}

{{citation

| last1 = Dress | first1 = Andreas W. M. | author1-link = Andreas Dress

| last2 = Yang | first2 = Lu

| last3 = Zeng | first3 = Zhenbing

| editor1-last = Du | editor1-first = Ding-Zhu

| editor2-last = Pardalos | editor2-first = Panos M.

| contribution = Heilbronn problem for six points in a planar convex body

| doi = 10.1007/978-1-4613-3557-3_13

| mr = 1376828

| pages = 173–190

| publisher = Kluwer Acad. Publ., Dordrecht

| series = Nonconvex Optim. Appl.

| title = Minimax and Applications

| volume = 4

| year = 1995| isbn = 978-1-4613-3559-7 }}

{{citation

| last = Goldberg | first = Michael

| journal = Mathematics Magazine

| doi = 10.2307/2687869

| issue = 3

| jstor = 2687869

| mr = 0296816

| pages = 135–144

| title = Maximizing the smallest triangle made by $n$ points in a square

| volume = 45

| year = 1972}}

{{citation

| last1 = Grimmett | first1 = G. | author1-link = Geoffrey Grimmett

| last2 = Janson | first2 = S. | author2-link = Svante Janson

| doi = 10.1002/rsa.10092

| issue = 2

| journal = Random Structures & Algorithms

| pages = 206–223

| s2cid = 12272636

| title = On smallest triangles

| volume = 23

| year = 2003}}

{{citation

| last1 = Jiang | first1 = Tao

| last2 = Li | first2 = Ming | author2-link = Ming Li

| last3 = Vitányi | first3 = Paul | author3-link = Paul Vitányi

| arxiv = math/9902043

| doi = 10.1002/rsa.10024

| issue = 2

| journal = Random Structures & Algorithms

| mr = 1884433

| pages = 206–219

| s2cid = 2079746

| title = The average-case area of Heilbronn-type triangles

| volume = 20

| year = 2002}}

{{citation

| last1 = Komlós | first1 = J. | author1-link = János Komlós (mathematician)

| last2 = Pintz | first2 = J. | author2-link = János Pintz

| last3 = Szemerédi | first3 = E. | author3-link = Endre Szemerédi

| doi = 10.1112/jlms/s2-24.3.385

| issue = 3

| journal = Journal of the London Mathematical Society

| mr = 0635870

| pages = 385–396

| title = On Heilbronn's triangle problem

| volume = 24

| year = 1981}}

{{citation

| last1 = Komlós | first1 = J. | author1-link = János Komlós (mathematician)

| last2 = Pintz | first2 = J. | author2-link = János Pintz

| last3 = Szemerédi | first3 = E. | author3-link = Endre Szemerédi

| doi = 10.1112/jlms/s2-25.1.13

| issue = 1

| journal = Journal of the London Mathematical Society

| mr = 0645860

| pages = 13–24

| title = A lower bound for Heilbronn's problem

| volume = 25

| year = 1982}}

{{citation

| last = Lefmann | first = Hanno

| doi = 10.1007/s00454-007-9041-y | doi-access = free

| issue = 3

| journal = Discrete & Computational Geometry

| mr = 2443292

| pages = 401–413

| title = Distributions of points in $d$ dimensions and large $k$-point simplices

| volume = 40

| year = 2008}}

{{citation

| last = Roth | first = K. F. | author-link = Klaus Roth

| doi = 10.1112/jlms/s1-26.3.198

| issue = 3

| journal = Journal of the London Mathematical Society

| pages = 198–204

| title = On a problem of Heilbronn

| volume = 26

| year = 1951}}

{{citation

| last1 = Yang | first1 = Lu

| last2 = Zeng | first2 = Zhenbing

| editor1-last = Du | editor1-first = Ding-Zhu

| editor2-last = Pardalos | editor2-first = Panos M.

| contribution = Heilbronn problem for seven points in a planar convex body

| doi = 10.1007/978-1-4613-3557-3_14

| mr = 1376829

| pages = 191–218

| publisher = Kluwer Acad. Publ., Dordrecht

| series = Nonconvex Optim. Appl.

| title = Minimax and Applications

| volume = 4

| year = 1995| isbn = 978-1-4613-3559-7

}}

{{citation

| last1 = Zeng | first1 = Zhenbing

| last2 = Chen | first2 = Liangyu

| editor1-last = Sturm | editor1-first = Thomas

| editor2-last = Zengler | editor2-first = Christoph

| contribution = On the Heilbronn optimal configuration of seven points in the square

| doi = 10.1007/978-3-642-21046-4_11

| location = Heidelberg

| mr = 2805061

| pages = 196–224

| publisher = Springer

| series = Lecture Notes in Computer Science

| title = Automated Deduction in Geometry: 7th International Workshop, ADG 2008, Shanghai, China, September 22-24, 2008, Revised Papers

| volume = 6301

| year = 2011| isbn = 978-3-642-21045-7

}}

{{cite arXiv |mode=cs2

| last1 = Cohen | first1 = Alex

| last2 = Pohoata | first2 = Cosmin

| last3 = Zakharov | first3 = Dmitrii

| eprint = 2305.18253

| title = A new upper bound for the Heilbronn triangle problem

| year = 2023| class = math.CO }}

{{citation

| first = Leila | last = Sloman

| date = September 8, 2023

| title = The Biggest Smallest Triangle Just Got Smaller

| url = https://www.quantamagazine.org/the-biggest-smallest-triangle-just-got-smaller-20230908

| access-date = September 9, 2023

| magazine = Quanta }}

}}

• {{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 $n$ 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