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How efficiently can one compute a polyhedral surface that is an \epsilon-approximation of a given triangulated surface in \R^3?
It is NP-hard to obtain the minimum-facet surface separating two nested convex polytopes [DG97], but polynomial-time approximation algorithms are known ([BG95], [MS95], [AS98]) for this case, and for separating two terrain surfaces, achieving factors within O(1) or O(\log n) of optimal. However, no polynomial-time approximation results are known for general surfaces.
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Joseph S. B. Mitchell and Subhash Suri. Separation and approximation of polyhedral objects. Comput. Geom. Theory Appl., 5:95–114, 1995.