Fast verification of convexity of piecewise-linear hypersurfaces

We show that a piecewise-linear (PL) complete immersion of a connected manifold of dimension n−1 into n-dimensional Euclidean space (n > 2) is the boundary of a convex polyhedron, bounded or unbounded, if and only if the interior of each (n− 3)-face has a point with a neighborhood (on the surface) that lies on the boundary of a convex body. No initial assumptions about the topology or orientability of the input surface are made, but if the surface is unbounded, the existence of a point of strict convexity is required. The theorem is derived from a refinement and generalization of Van Heijenoort’s (1952) theorem on locally convex manifolds to spherical spaces. Our convexity criterion for PL-manifolds implies an easy-to-implement polynomial-time algorithm for checking convexity of a given PL-hypersurface in n-dimensional Euclidean space. For n = 3 the number of arithmetic operations used by the algorithm is linear in the number of vertices; in the general dimension it is O(fn−2 n−3), where fn−2 n−3 is the number of incidences between (n − 2)and (n − 3)faces. The algorithm is optimal with respect to the highest degree of evaluated polynomial predicates. The algorithm works under significantly weaker assumptions and is easier to implement than convexity verification algorithms suggested by Mehlhorn et al. (1996a,b; 1999), and Devillers et al. (1998).