Unions of Fat Convex Polytopes Have Short Skeletons

The skeleton of a polyhedral set is the union of its edges and vertices. Let \(\mathcal {P}\) be a set of fat, convex polytopes in three dimensions with n vertices in total, and let f _max be the maximum complexity of any face of a polytope in \(\mathcal {P}\). We prove that the total length of the skeleton of the union of the polytopes in \(\mathcal {P}\) is at most O(α(n)?log^? n?logf _max) times the sum of the skeleton lengths of the individual polytopes.