Editing Heilbronn triangle problem (section)

==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}}}}