On locally convex PL-manifolds and fast verification of convexity

We show that a realization of a closed connected PL-manifold of dimension n-1 in Euclidean n-space (n>2) is the boundary of a convex polyhedron if and only if the interior of each (n-3)-face has a point, which has a neighborhood lying on the boundary of a convex n-dimensional body. This result is derived from a generalization of Van Heijenoort's theorem on locally convex manifolds to spherical spaces. We also give a brief analysis of how local convexity and topology of non-compact surfaces are related to global convexity in the hyperbolic space. Our convexity criterion for PL-manifolds imply an easy polynomial-time algorithm for checking convexity of a given closed compact PL-surface in Euclidean of spherical space of dimension n>2.