172,157 research outputs found

    The Stretch Factor of the Delaunay Triangulation Is Less Than 1.998

    Full text link
    Let SS be a finite set of points in the Euclidean plane. Let DD be a Delaunay triangulation of SS. The {\em stretch factor} (also known as {\em dilation} or {\em spanning ratio}) of DD is the maximum ratio, among all points pp and qq in SS, of the shortest path distance from pp to qq in DD over the Euclidean distance pq||pq||. Proving a tight bound on the stretch factor of the Delaunay triangulation has been a long standing open problem in computational geometry. In this paper we prove that the stretch factor of the Delaunay triangulation of a set of points in the plane is less than ρ=1.998\rho = 1.998, improving the previous best upper bound of 2.42 by Keil and Gutwin (1989). Our bound 1.998 is better than the current upper bound of 2.33 for the special case when the point set is in convex position by Cui, Kanj and Xia (2009). This upper bound breaks the barrier 2, which is significant because previously no family of plane graphs was known to have a stretch factor guaranteed to be less than 2 on any set of points.Comment: 41 pages, 16 figures. A preliminary version of this paper appeared in the Proceedings of the 27th Annual Symposium on Computational Geometry (SoCG 2011). This is a revised version of the previous preprint [v1

    On submanifolds whose tubular hypersurfaces have constant mean curvatures

    Full text link
    Motivated by the theory of isoparametric hypersurfaces, we study submanifolds whose tubular hypersurfaces have some constant "higher order mean curvatures". Here a kk-th order mean curvature QkQ_k (k1k\geq1) of a hypersurface MnM^n is defined as the kk-th power sum of the principal curvatures, or equivalently, of the shape operator. Many necessary restrictions involving principal curvatures, higher order mean curvatures and Jacobi operators on such submanifolds are obtained, which, among other things, generalize some classical results in the theory of isoparametric hypersurfaces given by E. Cartan, K. Nomizu, H. F. M{\"u}nzner, Q. M. Wang, \emph{etc.}. As an application, we finally get a geometrical filtration for the focal varieties of isoparametric functions on a complete Riemannian manifold.Comment: 29 page