Fast Verification of Convexity of Piecewise-linear Surfaces

We show that a PL-realization of a closed connected manifold of dimension n − 1 in R (n ≥ 3) 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 the spherical case. No initial assumptions about the topology or orientability of the input surface are made. Our convexity criterion for PL-manifolds implies an easy polynomial-time algorithm for checking convexity of a given PL-surface in R. 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. There is a number of theorems that infer global convexity from local convexity. The oldest one belongs to Jacque Hadamard (1897) and asserts that any compact smooth surface embedded in R, with strictly positive Gaussian curvature, is the boundary of a convex body. Local convexity can be defined in many different ways (see van Heijenoort (1952) for a survey). We will use Bouligand’s (1932) notion of local convexity. In this definition a surface M in the affine space R is called locally convex at point p if p has a neighborhood which lies on the boundary of a convex n-dimensional body Kp; if Kp\p lies in an open half-space defined by a hyperplane containing p, M is called strictly convex at p. This paper is mainly devoted to local convexity of piecewise-linear (PL) surfaces, in particular, polytopes. A PL-surface in R is a pair M = (M, r), where M is a topological manifold with a fixed cell-partition and r is a continuous realization map from M to R that satisfies the following conditions: 1) r is a bijection on the closures of all cells of M 1) for each k-cell C of M the image r(C) lies on a k-dimensional affine subspace of R; r(C) is then called a k-face of M .