Reconstructing orthogonal polyhedra from putative vertex sets

In this paper we study the problem of reconstructing orthogonal polyhedra from a putative vertex set, i.e., we are given a set of points and want to find an orthogonal polyhedron for which this is the set of vertices. This is well-studied in 2D; we mostly focus on 3D, and on the case where the given set of points may be rotated beforehand. We obtain fast algorithms for reconstruction in the case where the answer must be orthogonally convex.