Suppose $p_1,\ldots,p_n$ are points in $\mathbb{R}^3$, and suppose $\delta_1,\ldots,\delta_n$ are positive real numbers, each less than $2\pi$, whose sum is $4\pi$. Is there a polyhedron $\mathcal P$ which has a vertex at $p_i$ of angular defect $\delta_i$ (meaning that the sum of the interior angles of the faces meeting at $p_i$ is $2\pi-\delta_i$) for each $i=1,\ldots, n$, and possibly some extra vertices all of whose angular defect $0$? That is, I am specifying some of the vertices and angular defects, but allowing the addition of extra vertices on the condition that those vertices have angular defect zero.
If so, is there a constructive proof? If not, are there known constraints on the points and/or defects?
The question can be extended more broadly: the condition that the defects are positive could be dropped and we could ask that $\delta_1+\cdots+\delta_n=2\pi \chi$.
