I am wondering if there are any connection between convex polygon and castable object? What can we say about castability of the object if we know that the object is convex polygon and vice versa.

Let's gather together few basic things that we have to know.

The object is castable if it can removed from the mold.

The polyhedron P can be removed from its mold by a translation in direction $\vec{d}$ if and only if $\vec{d}$ makes an angle of at least $90^{\circ}$ with the outward normal of all ordinary facets of P.

For a arbitrary object testing for castability has time complexity $O(n^2)$. In my opinion, for a convex polygon if could be improved to linear time, because for every new top facet we should test that the vector $\vec{d}$ makes an angle at least $90^{\circ}$ with outward normal not of all but only of two adjacent ordinary facets of P.

If this is true at least we have improvement in testing for castability in case of convex polygon.

We else can we state about castability and convexity. Especially interesting to know, if castability tells us something about convexity.