Adjacency method for extreme Delaunay polytopes

Perfect Delaunay Polytopes and Perfect Quadratic Functions on Lattices

A polytope $D$ whose vertices belong to a lattice of rank $d$ is Delaunay if there is a circumscribing $d$-dimensional ellipsoid, $E$, with interior free of lattice points so that the vertices of $D$ lie on $E$. If in addition, the ellipsoid $E$ is uniquely determined by $D$, we call $D$ perfect. That is, a perfect Delaunay polytope is a lattice polytope with a circumscribing empty ellipsoid $E$, where the quadratic surface $\partial E$ both contains the vertices of $D$ and is determined by them. We have been able to construct infinite sequences of perfect Delaunay polytopes, one perfect polyt…

Universality theorems for inscribed polytopes and Delaunay triangulations

We prove that every primary basic semialgebraic set is homotopy equivalent to the set of inscribed realizations (up to Möbius transformation) of a polytope. If the semialgebraic set is moreover open, then, in addition, we prove that (up to homotopy) it is a retract of the realization space of some inscribed neighborly (and simplicial) polytope. We also show that all algebraic extensions of $\mathbb{Q}$ are needed to coordinatize inscribed polytopes. These statements show that inscribed polytopes exhibit the Mnëv universality phenomenon. Via stereographic projections, these theorems have a dire…

Alexandrov's theorem, weighted Delaunay triangulations, and mixed volumes

We present a constructive proof of Alexandrov's theorem regarding the existence of a convex polytope with a given metric on the boundary. The polytope is obtained as a result of a certain deformation in the class of generalized convex polytopes with the given boundary. We study the space of generalized convex polytopes and discover a relation with the weighted Delaunay triangulations of polyhedral surfaces. The existence of the deformation follows from the non-degeneracy of the Hessian of the total scalar curvature of a positively curved generalized convex polytope. The latter is shown to be e…

The hypermetric cone on seven vertices

The hypermetric cone $HYP_n$ is the set of vectors $(d_{ij})_{1\leq i< j\leq n}$ satisfying the inequalities $\sum_{1\leq i<j\leq n} b_ib_jd_{ij}\leq 0 with b_i\in\Z and \sum_{i=1}^{n}b_i=1$. A Delaunay polytope of a lattice is called extremal if the only affine bijective transformations of it into a Delaunay polytope, are the homotheties; there is a correspondance between such Delaunay polytopes and extreme rays of $HYP_n$. We show that unique Delaunay polytopes of root lattice $A_1$ and $E_6$ are the only extreme Delaunay polytopes of dimension at most 6. We describe also the skeletons and a…

Perfect Delaunay Polytopes and Perfect Inhomogeneous Forms

A lattice Delaunay polytope D is called perfect if it has the property that there is a unique circumscribing ellipsoid with interior free of lattice points, and with the surface containing only those lattice points that are the vertices of D. An inhomogeneous quadratic form is called perfect if it is determined by such a circumscribing ''empty ellipsoid'' uniquely up to a scale factor. Perfect inhomogeneous forms are associated with perfect Delaunay polytopes in much the way that perfect homogeneous forms are associated with perfect point lattices. We have been able to construct some infinite …

Random Inscribed Polytopes Have Similar Radius Functions as Poisson-Delaunay Mosaics

Using the geodesic distance on the $n$-dimensional sphere, we study the expected radius function of the Delaunay mosaic of a random set of points. Specifically, we consider the partition of the mosaic into intervals of the radius function and determine the expected number of intervals whose radii are less than or equal to a given threshold. Assuming the points are not contained in a hemisphere, the Delaunay mosaic is isomorphic to the boundary complex of the convex hull in $\mathbb{R}^{n+1}$, so we also get the expected number of faces of a random inscribed polytope. We find that the expectati…

A New Algorithm in Geometry of Numbers

A lattice Delaunay polytope P is called perfect if its Delaunay sphere is the only ellipsoid circumscribed about P. We present a new algorithm for finding perfect Delaunay polytopes. Our method overcomes the major shortcomings of the previously used method. We have implemented and used our algorithm for finding perfect Delaunay polytopes in dimensions 6, 7, 8. Our findings lead to a new conjecture that sheds light on the structure of lattice Delaunay tilings.

Construction of the Voronoi Diagram and Secondary Polytope

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…

Explicit Construction of the Voronoi and Delaunay Cells of W(An) and W(Dn) Lattices and Their Facets

Voronoi and Delaunay (Delone) cells of the root and weight lattices of the Coxeter-Weyl groups W(an) and W(dn) are constructed. The face centered cubic (fcc) and body centered cubic (bcc)lattices are obtained in this context. Basic definitions are introduced such as parallelotope, fundamental simplex, contact polytope, root polytope, Voronoi cell, Delone cell, n-simplex, n-octahedron (cross polytope), n-cube and n-hemicube and their volumes are calculated. Voronoi cell of the root lattice is constructed as the dual of the root polytope which turns out to be the union of Delone cells. It is sho…

Computational methods for cones and polytopes with symmetry

Every polyhedral cone can be described either by its facets or by its extreme rays. Computation of one description from the other is a problem that can be very complex, i.e. one encounter the combinatorial explosion. We present here several methods which allow us to use symmetries for the computation of extreme rays and facets of cones. I give an algorithm for computing adjacencies using a linear programming approach. I give a complete description of the method of incidence, adjacency decomposition method and of the subcone method used for computing with symmetric cones.

The decomposition of the hypermetric cone into L-domains

The hypermetric cone $\HYP_{n+1}$ is the parameter space of basic Delaunay polytopes in n-dimensional lattice. The cone $\HYP_{n+1}$ is polyhedral; one way of seeing this is that modulo image by the covariance map $\HYP_{n+1}$ is a finite union of L-domains, i.e., of parameter space of full Delaunay tessellations. In this paper, we study this partition of the hypermetric cone into L-domains. In particular, it is proved that the cone $\HYP_{n+1}$ of hypermetrics on n+1 points contains exactly {1/2}n! principal L-domains. We give a detailed description of the decomposition of $\HYP_{n+1}$ for n=…

可内接扇 I：内接锥与虚拟多胞形

Inscribable fans I: Inscribed cones and virtual polytopes

本文研究内接于球面的多胞形，引入内接锥与内接虚拟多胞形，证明其封闭性及多项式时间可判定性，并分类二维可内接扇及若干多胞形类，关联反射群胚。