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### 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

### On submanifolds whose tubular hypersurfaces have constant mean curvatures

Motivated by the theory of isoparametric hypersurfaces, we study submanifolds
whose tubular hypersurfaces have some constant "higher order mean curvatures".
Here a $k$-th order mean curvature $Q_k$ ($k\geq1$) of a hypersurface $M^n$ is
defined as the $k$-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

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