Give a polygon $P$ in $\mathbb{R}^2$, the centroid of $P$ is $(v_1+\cdots+v_n)/n$, where $v_1,\ldots,v_n\in\mathbb{R}^2$ are the vertices of $P$.
Suppose $P$ and $Q$ are two polygons in $\mathbb{R}^2$ satisfies:
- They have the same centroid.
- The ratios of their shaows in every direction are at most $C$, i.e. for all line $l$ going through their centroid, $$length(l\cap P)/length(l\cap Q)\leq C.$$
Claim: The ratios of their areas is at most $C^2$.
Is the claim correct?
I don't think this can be true.
Consider this figure:
Although this has circular segments, it can be approximated by a polygon. Cuts through its centroid are almost all the same length, and if I did my calculations right, its area is $\frac{10}{9}$ times that of a circle with that diameter.
So here are two figures, which when approximated by polygons $P$ and $Q$, will have almost the same cut lengths through their centroid, but a distinctly different area.
If your polygons approximate them closely enough, $C$ will be close to $1$, and for sure you can get $C^2<\frac{10}{9}$, and construct a counterexample to your claim.
I think that even with convex polygons, a similar technique will result in a counterexample.