Suppose you have a polyhedra, not convex, and that the number of vertices is $N$. Furthermore, suppose that there are $O(N)$ edges.
Using the Euler Characteristic, $\chi$ , we have,
$F = E-V+ \chi = O(N) + \chi$
So, does there exist an upper bound on $\chi$ such that there are $O(N)$ faces?