If I have $n, 1 < i < n, $ surfaces composed of $f_i$ faces and $v_i$ vertices, how would I go about finding the average surface?
(I'm unsure what I mean by average - intuitively it's obvious, but mathematically I am not sure how to express it...)
If I have $n, 1 < i < n, $ surfaces composed of $f_i$ faces and $v_i$ vertices, how would I go about finding the average surface?
(I'm unsure what I mean by average - intuitively it's obvious, but mathematically I am not sure how to express it...)
I too am unsure what you mean by "average", but any sensible response to your question should surely involve the recent work of Babson, Hoffman, and Kahle on random 2-complexes (there's a sharp threshold for simple connectedness! It's cool!) and the works cited therein by Linial-Meshulam and Meshulam-Wallach.
Your question is quite underspecified, and so difficult to answer. Let me narrow it considerably, perhaps well beneath your interests, to convex polyhedra. Then you can use this result: The convex combination of convex sets is itself convex.
But probably you are interested in nonconvex polyhedra? You might be able to proceed by partitioning your nonconvex polyhedron into convex polyhedra, an interesting challenge in its own right (see the 2D equivalent discussed on MO here: "Partitioning a polygon into convex parts").
And as Douglas' question indicated, you might be clearer on whether all your polyhedra have the same genus.