The Stretch Factor of the Delaunay Triangulation Is Less Than 1.998

Let $S$ be a finite set of points in the Euclidean plane. Let $D$ be a Delaunay triangulation of $S$. The {\em stretch factor} (also known as {\em dilation} or {\em spanning ratio}) of $D$ is the maximum ratio, among all points $p$ and $q$ in $S$, of the shortest path distance from $p$ to $q$ in $D$ over the Euclidean distance $||pq||$. 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 $\rho = 1.998$, 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

Comparison theorems for manifolds with mean convex boundary

Let M-n be an n-dimensional Riemannian manifold with boundary partial derivative M. Assuming that Ricci curvature is bounded from below by (n - 1)k, for k is an element of R, we give a sharp estimate of the upper bound of rho(x) = d(x, partial derivative M), in terms of the mean curvature bound of the boundary. When partial derivative M is compact, the upper bound is achieved if and only if M is isometric to a disk in space form. A Kahler version of estimation is also proved. Moreover, we prove a Laplacian comparison theorem for distance function to the boundary of Kahler manifold and also estimate the first eigenvalue of the real Laplacian.SCI(E)[email protected]