Unions of Fat Convex Polytopes Have Short Skeletons
Boris S. Aronov, Mark de Berg · Discrete & Computational Geometry · 2012
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.