Let $G$ be the edge graph of a convex subdivision of a convex polygon $P$ in the plane. I would like to construct a convex polyhedral cap $C$ (with zero boundary values) over $P$ whose edges project onto $G$ and whose interior vertices have prescribed curvature.

It is well known that for $C$ to satisfy the edge condition, $G$ must meet certain requirements in the context of *Maxwell-Cremona correspondence*; see the answer to this earlier question. So let us assume that $G$ meets these conditions.

It is also known that one can construct a convex cap over $P$ whose interior vertices project onto those of $G$ and have prescribed curvature values as long as these values are all positive and add up to less than $2\pi$. This is a lemma of Pogorelov; see p. 319 of Pak's Lecture Notes.

In short, one can prescribe the edges of $C$, and one can also prescribe the curvature of the interior vertices of $C$ independently, but to what extent can one do both at the same time:

Question:Suppose that we have a convex subdivision of a convex polygon which can be lifted to a convex cap. What are all the possible values for the curvatures at the interior vertices of this cap?

I know that theoretically one can compute these curvatures in a given situation via the Maxwell-Cremona machinery; however, I am wondering if there are some relatively nice integrability conditions which can be formulated in general, to somehow yield a more refined version of the Maxwell-Cremona correspondence.