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 $\frac{S^{\prime}}{S}$.
$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $
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.

Not a complete answer, and surely not elegant, but here's an approach.
The idea is to consider "polygonal conics" in the following sense. Take the cone $K$ given by
\begin{equation}
K:\qquad x^2+y^2=z^2
\end{equation}
and slice it with the plane $z=1$.
We get a circle $\gamma$, and let's inscribe in this circle a regular pentagon, $P$ (The idea actually works with a regular $n$-gon with $n=2k+1$... should it be called $n$-oddgon?). The diagonals of this pentagon define a smaller pentagon $P'$. Let also $\gamma'$ be the circle in which $P'$ is itself inscribed in.
Let also $C,C',S,S'$ be the areas of $\gamma,\gamma',P,P'$ respectively. We have
\begin{equation}
\frac{C'}{C}=\frac{S'}{S}
\end{equation}
since all these areas depend only on the radiuses of $\gamma$ and $\gamma'$.
Let's now slice the initial cone and the cone $K'$ generated by the circle $\gamma'$ with a general plane
\begin{equation}
(x-x_0)\cos\theta+(z-z_0)\sin\theta=0
\end{equation}
(Informally, it's the plane passing through $(x_0,0,z_0)$ and its normal vector lies in the plane $y=0$ and makes an angle $\theta$ with the $x$-axis).
We get two nested conics. Considering only the cases when these conics are ellipses, we can compute the ratio of their areas (area of the inner ellipse over area of the outer one) and one finds that the maximum of this value occurs when both are circles.
When only the inner conic is an ellipse, the ratio is zero.
Considering now the "pentagonal cones" obtained by extruding the original nested pentagons along the two cones $K$ and $K'$ and reasoning in a similar way as before, we have that the ratio between the areas of the nested "pentagonal conics" has a maximum when the pentagons are regular.
A problem with this approach is that I don't have a clue of what class of pentagons we are dealing with, as it may be too narrow to be of any real interest for the problem.
UPDATE: in the case of a pentagon, since five points in general position determine a conic, I guess we have covered all the possible cases.
