Determining the Heilbronn Configuration of Seven Points in Triangles via Symbolic Computation

In this paper we first recall some rigorously proved results related to the Heilbronn numbers and the corresponding optimal configurations of n = 5, 6, 7 points in squares, disks, and general convex bodies K in the plane, n = 5, 6 points in triangles and a bundle of approximate results obtained by numeric computation in the Introduction section. And then in the second section we will present a proof to a conjecture on the Heilbronn number for seven points in the triangle through solving a group of non-linear optimization problems via symbolic computation. In the third section we list three unsolved well-formed such non-linear programming problems corresponding to Heilbronn configurations for n = 8, 9 points in squares and 8 points in triangle, we expect they can be solved by similar method we used in the Section two. In the final section we mention two generalizations of the classic Heilbronn triangle problem. The paper aims to provide a concise guide to further studies on Heilbronn-type problems for small number of points in specific convex bodies.