Editing Heilbronn triangle problem (section)

==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}}}}