On locally convex PL-manifolds and fast verification of convexity

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. Our convexity criterion for PL-manifolds implies an easy polynomial-time algorithm for checking convexity of a given PL-surface in R. 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 . Thus, r need not be an immersion, but its restriction to the closure of any cell of M must be. By a fixed cell-partition of M we mean that M has a structure of a CW-complex where all gluing mappings are homeomorphisms (such complexes are called regular by J.H.C. Whitehead). All cells and faces are assumed to be open. We will also call M = (M, r) a PL-realization of M in R.