inversely proportional One way to see this is to triangulate the convex hull of the given point and choose the smallest of the triangles in the triangulation. Another is to sort the points by their and to choose the three consecutive points in this ordering whose are the closest together. In the first paper published on the Heilbronn triangle problem, in 1951, Klaus Roth proved a stronger upper bound of the form The best bound known to date is of the form for some proven by . Specific shapes and numbers has investigated the optimal arrangements of points in a square, for up to 16. 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 improved Goldberg's bounds, and for these values the solutions include points interior to the square. 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. The following are the best known solutions for 7–12 points in a unit square, found through simulated annealing; the arrangement for seven points is known to be optimal. 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 with area one, the regular hexagon is the one that for this shape, with six points optimally placed at the hexagon vertices. The convex shapes of unit area that maximize have 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. Variations involving the volume of higher-dimensional simplices have also been studied. Rather than considering simplices, another higher-dimensional version adds another and asks for placements of points in the unit hypercube that maximize the minimum volume of the convex hull of