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Finite or discrete collections of geometric objects. Packings, tilings, polyhedra, polytopes, intersection, arrangements, rigidity.
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If a polyhedron in $\mathbb{R}^3$ has local intersections, does it also have more global int...
Sorry, my previous answer is based on the impression that the "manifold" can mean "manifold with boundary. In that case, the countercase can be constructed easily (see my previous answer).
But for tru …