# Minmax problem for polygons

Let $\text{Pol}_n$ be the set of all convex polygons on a plane with $n$ vertices. For $P\in \text{Pol}_n$ denote by $\text{Tr}(P)$ the set of all triangles which vertices are some vertices of $P$. I want to find an explicit formula for the function $$\Phi(n)=\inf\limits_{P\in \text{Pol}_n}\max\limits_{T\in \text{Tr}(p)}\frac{\text{area}(T)}{\text{area}(P)}$$ It is not hard to prove that $\Phi(3)=1$, $\Phi(4)=1/2$. For $n\geq 5$ we have an estimation $\Phi(n)\geq 1/(n-2)$.

Here you can find some attempts to solve it. Any ideas are appreciated!

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Somewhat related question: mathoverflow.net/questions/34865/… – Gerry Myerson Mar 11 '12 at 1:18
An algorithm for finding a maximal triangle is given in cs.bgu.ac.il/~sityon/soda06.pdf. I don't know whether the paper, or its references, contains any theoretical results on the size of said triangle. – Gerry Myerson Mar 11 '12 at 1:21