## Heilbronn problem for seven points in a planar body |

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1/9 is best ACADEMIA SINICA affine coordinate system affine regular n-gon b)uv best possible Bielefeld bounds of Hn(K CHENGDU Choose an affine coefficient completes the proof compute configuration of seven contradicts convex hull convex pentagon convex polygon easy to calculate easy to check easy to know easy to observe easy to verify edge equation system feasible set Given a convex going to prove Heilbronn arrangement Heilbronn numbers hence hexagon ie easy implicit function incident with r7 largest root Let r'2 let r3 loose relative main theorem minimum point Open Problems peripheral triangles planar convex body Proposition 2.1 prove the main put r3 quadrilateral r2r3r4 is loose r3re r7 G rir2r3 r7 is contained r7 is stable rirjrk Smallest Triangle Solving the equation stability of rj three diagonal tight triangles transformed the problem unique critical point upper bound uv s.t. vertices view of Lemma