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 fmax 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⋅logfmax) times the sum of the skeleton lengths of the individual polytopes.

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