## Minmax problem for polygons

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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 2012 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 2012 at 1:21
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