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Comment: 9 pages, (v2) several small changes and corrections suggested by referees, accepted in Journal of Combinatorial Theory, Series A
Comment: Several changes in the last section. In the original version of this paper we claimed that any regular triangulation of a convex d-polytope has at least d ears. For a proof we used the same arguments as in Schatteman's paper [22]. Since this paper has certain gaps (see our paper [1]), the d -ears problem of a regular triangulation is still open
Comment: 10 pages
Comment: Talk at the Fifth European Congress of Mathematics, Amsterdam,14-18 July 2008
Comment: 30 pages, 9 figures
We defined several functionals on the set of all triangulations of the finite system of points in d-space achieving global minimum on the Delaunay triangulation (DT). We consider a so called "parabolic" functional and prove it attains its minimum on DT in all dimensions. As the second example we treat "mean radius" functional (mean of circumcircle radii of triangles) for planar triangulations. As the third example we treat a so called "harmonic" functional. For a triangle this functional equals the sum of squares of sides over area. Finally, we consider a discrete analog of the Dirichlet functional. DT is optimal for these functionals only in dimension two.
A set S of n points in general position in R^d defines the unique Voronoi diagram of S. Its dual tessellation is the Delaunay triangulation (DT) of S. In this paper we consider the parabolic functional on the set of triangulations of S and prove that it attains its minimum at DT in all dimensions. The Delaunay triangulation of S is corresponding to a vertex of the secondary polytope of S. We proposed an algorithm for DT's construction, where the parabolic functional and the secondary polytope are used. Finally, we considered a discrete analog of the Dirichlet functional. DT is optimal for this functional only in two dimensions.
Comment: 10 pages, 5 figures
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