A Heilbronn Type Inequality for Plane Nonagons

Zhenbing Zeng, Jian Lu, Lydia Dehbi, Liangyu Chen, Jianlin Wang

Abstract: In this paper, we present a proof of the property that for any convex nonagon $$P_1P_2\ldots P_9$$ in the plane, the smallest area of a triangle $$P_{i}P_{j}P_{k} (1\le i< j < k \le 9)$$ is at most a fraction of $$4\cdot \sin ^2(\pi /9)/9= 0.05199\ldots $$ of the area of the nonagon. The problems is transformed into an optimization problem with bilinear constraints and solved by symbolic computation with Maple.