About the ratio of the areas of a convex pentagon and the inner pentagon made by the five diagonals I've thought about the following question for a month, but I'm facing difficulty. 
Question : Letting $S{^\prime}$ be the area of the inner pentagon made by the five diagonals of a convex pentagon whose area is $S$, then find the max of $\frac{S^{\prime}}{S}$.
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It seems that a regular pentagon and its affine images would give the max. However, I don't have any good idea without tedious calculations. Can anyone help?
Update : I crossposted to MO.

The area ratio $S’/S$ is always $k^2$ where $$k = {1-\sin(\pi/10) \over 1 + \cos(\pi/5)}.$$ $k$ is the ratio of the perimeter of the inner pentagon to the outer pentagon. 
To see this, note that the inner pentagon appears to to be similar the outer pentagon but rotated by 180 degrees. Assume the outer pentagon is inscribed in a unit circle. Label the outer regular pentagon vertices as ABCDE with A at the top. Label the inner pentagon vertices as abcde but starting with a at the bottom. The triangles ACD and Acd are similar. It’s easily seen that the altitude of the Acd triangle is $1-\sin(\pi/10)$ and that of the ACD triangle is $1 + \cos(\pi/5)$. Thus, the ratio of cd to CD is $k$. Since cd and CD are sides of regular pentagons, the perimeter ratio must be $k$ and the area ratio must be $k^2$.
I leave the proof to others, but I assert that the following are invariant under any affine transformation:


*

*parallel lines remain parallel;

*the ratio of line sub-segments (P,X) and (X,Q) is constant if line
segment (P,Q) is intersected at X by another line segment (p,q);

*similar polygons remain similar; and, as a result,

*the perimeter and area ratios of similar polygons are constant.


The results have been checked with a MATLAB script that generates random affine transformations. Might add that these results should hold for other regular polygons except for the value of $k$. For example, $k = 1/3$ for a hexagon.
