In discrete geometry and discrepancy theory, the Heilbronn triangle problem is a problem of placing points in the plane, avoiding triangles of small area. It is named after Hans Heilbronn, who conjectured that, no matter how points are placed in a given area, the smallest triangle area will be at most inversely proportional to the square of the number of points. His conjecture was proven false, but the asymptotic growth rate of the minimum triangle area remains unknown. Definition The Heilbronn triangle problem concerns the placement of points within a shape in the plane, such as the unit square or the unit disk, for a given 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 points be placed to maximize the area of the smallest More formally, the shape may be assumed to be a compact set 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, 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 points, rather than only a sequence of placements approaching optimality. The number may be defined as the area of the smallest triangle in this optimal An example is shown in the figure, with six points in a unit square. These six points form 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, Although researchers have studied the value of for specific shapes and specific small numbers of points, Heilbronn was concerned instead about its asymptotic behavior: if the shape is held fixed, but varies, how does the area of the smallest