OK, maybe then a better assumption to rule out the trivial or degenerate cases would be 3-connected. I have now edited the question accordingly. Thanks.

In the degenerate case there are really no edges. I edited the question to specifically stipulate the existence of an interior vertex in the subdivision, so that the degenerate case does not arise. Thanks for pointing this out.

In that case the function would be zero and the cap would be degenerate. The question is interesting only when the partition has some vertices in the interior of the polygon.

We can think of the cap as the graph of a piecewise linear concave function over P with zero boundary values. So it looks like a dome over P. Further the cap is a topological disk, so its boundary is just the topological boundary of the disk.

Thank you! I will look up the references. In the meantime I wonder: are there necessary an sufficient conditions known for the existence of a lifting of the subdivision to a convex cap? If so, are there explicit procedures for constructing the cap?