Fast Verification of Convexity of Piecewise-linear Surfaces

Abstract

We show that a realization of a closed connected PL-manifold of dimension n-1 in n-dimensional Euclidean space (n>2) is the boundary of a convex polyhedron (finite or infinite) if and only if the interior of each (n-3)-face has a point, which has a neighborhood lying on the boundary of an n-dimensional convex body. No initial assumptions about the topology or orientability of the input surface are made. The theorem is derived from a refinement and generalization of Van Heijenoort's theorem on locally convex manifolds to spherical spaces. Our convexity criterion for PL-manifolds implies an easy polynomial-time algorithm for checking convexity of a given PL-surface in n-dimensional Euclidean or spherical space, n>2. The algorithm is worst case optimal with respect to both the number of operations and the algebraic degree. The algorithm works under significantly weaker assumptions and is easier to implement than convexity verification algorithms suggested by Mehlhorn et al (1996-1999), and Devillers et al.(1998). A paradigm of approximate convexity is suggested and a simplified algorithm of smaller degree and complexity is suggested for approximate floating point convexity verification.