Given a non-degenerate non-self-intersecting polyhedron $P$, consider the average of the dihedral angles at each edge in $P$. For small polyhedra, this average can be fairly small; for instance, a regular tetrahedron has an average dihedral angle of $70.53^\circ$, and the tetrahedron with vertices at $(1,0,\epsilon),(-1,0,\epsilon),(0,1,-\epsilon),(0,-1,-\epsilon)$ has an average dihedral angle approaching $60^\circ$.
I am interested in the limiting value of this average for large polyhedra, i.e. as the number of edges, faces, and vertices go to infinity. (If any one of these measures goes to infinity, so do the others.)
By taking an $N$-gonal pyramid with the vertex extremely close to the base, one can get an average angle approaching $90^\circ$ (half the dihedral angles are negligible, half are extremely close to $180^\circ$). Is it possible to get any smaller than this in the limit?
So far as I can tell, it seems like no polyhedron of any size can have an average dihedral angle less than $60^\circ$ without being self-intersecting; a proof or counterexample to this assertion would be welcome.
It is possible to get at least as low as $\arccos(\sqrt{5}/3)$, or about $41.8$ degrees, since that is the average dihedral angle in a great icosahedron.