Delaunay Supertopes
نویسندگان
چکیده
A perfect inhomogeneous quadratic form is defined by that it can be reconstructed in a unique way from its arithmetic minimum and all integral vectors on which it is attained. Such a form defines an integral Delaunay polytope that has a remarkable property of having a unique circumscribed ellipsoid whose interior is free of integral points. A polytope with this uniqueness property is called a perfect Delaunay polytope, or Delaunay supertope. We present an explicit construction of an infinite series (in two integer parameters) of infinite sequences (in dimension) of Delaunay supertopes. This paper is an extended announcement of our results. Complete proofs and history of the subject will be given in a subsequent journal paper.
منابع مشابه
The Stability of Delaunay Triangulations
We introduce a parametrized notion of genericity for Delaunay triangulations which, in particular, implies that the Delaunay simplices of δ-generic point sets are thick. Equipped with this notion, we study the stability of Delaunay triangulations under perturbations of the metric and of the vertex positions. We quantify the magnitude of the perturbations under which the Delaunay triangulation r...
متن کاملDelaunay Cells for Arrangements of Flats in Hyperbolic Space
For n + 1 disjoint flats of dimension k in H, we produce a Delaunay cell which is a generalization of the Delaunay simplex associated to n + 1 points in H. Combinatorially, these Delaunay cells resemble truncated n-dimensional simplices. For certain classes of arrangements of flats in H, we prove that these Delaunay cells can be glued together to form a Delaunay complex, with the result that al...
متن کاملConstructing Intrinsic Delaunay Triangulations of Submanifolds
We describe an algorithm to construct an intrinsic Delaunay triangulation of a smooth closed submanifold of Euclidean space. Using results established in a companion paper on the stability of Delaunay triangulations on δ-generic point sets, we establish sampling criteria which ensure that the intrinsic Delaunay complex coincides with the restricted Delaunay complex and also with the recently in...
متن کاملMinimal Set of Constraints for 2D Constrained Delaunay Reconstruction
Given a triangulation T of n points in the plane, we are interested in the minimal set of edges in T such that T can be reconstructed from this set (and the vertices of T ) using constrained Delaunay triangulation. We show that this minimal set consists of the non locally Delaunay edges of T , and that its cardinality is less than or equal to n+ i=2 (if i is the number of interior points in T )...
متن کاملOn Delaunay Oriented Matroids for Convex Distance Functions
For any nite point set S in Ed, an oriented matroid DOM(S) can be de ned in terms of how S is partitioned by Euclidean hyperspheres. This oriented matroid is related to the Delaunay triangulation of S and is realizable, because of the lifting property of Delaunay triangulations. We prove that the same construction of a Delaunay oriented matroid can be performed with respect to any smooth, stric...
متن کامل