arXiv subjects

Konstantin Rybnikov

Publications and source records attributed to Konstantin Rybnikov.

15 recordsLinked to original sources

arXiv2009

Perfect but not generating Delaunay polytopes

In his seminal 1951 paper "Extreme forms" Coxeter \cite{cox51} observed that for $n \ge 9$ one can add vectors to the perfect lattice $\sfA_9$ so that the resulting perfect lattice, called $\sfA_9^2$ by Coxeter, has exactly the same set of minimal vectors. An inhomogeneous analog of the notion of perfect lattice is that of a lattice with a perfect Delaunay polytope: the vertices of a perfect Delaunay polytope are the analogs of minimal vectors in a perfect lattice. We find a new infinite series $P(n,s)$ for $s\geq 2$ and $n+1\geq 4s$ of $n$-dimensional perfect Delaunay polytopes. A remarkable property of this series is that for certain values of $s$ and all $n \ge 13$ one can add points to the integer affine span of $P(n,s)$ in such a way that $P(n,s)$ remains a perfect Delaunay polytope in the new lattice. Thus, we have constructed an inhomogeneous analog of the remarkable relationship between $\sfA_9$ and $\sfA_9^2$.

math.CO↗

arXiv2009

Delaunay polytopes derived from the Leech lattice

Given a lattice L of R^n, a polytope D is called a Delaunay polytope in L if the set of its vertices is S\cap L where S is a sphere having no lattice points in its interior. D is called perfect if the only ellipsoid in R^n that contains S\cap L is exactly S. For a vector v of the Leech lattice Λ_{24} we define Λ_{24}(v) to be the lattice of vectors of Λ_{24} orthogonal to v. We studied Delaunay polytopes of L=Λ_{24}(v) for |v|^2<=22. We found some remarkable examples of Delaunay polytopes in such lattices and disproved a number of long standing conjectures. In particular, we discovered: --Perfect Delaunay polytopes of lattice width 4; previously, the largest known width was 2. --Perfect Delaunay polytopes in L, which can be extended to perfect Delaunay polytopes in superlattices of L of the same dimension. --Polytopes that are perfect Delaunay with respect to two lattices $L\subset L'$ of the same dimension. --Perfect Delaunay polytopes D for L with |Aut L|=6|Aut D|: all previously known examples had |Aut L|=|Aut D| or |Aut L|=2|Aut D|. --Antisymmetric perfect Delaunay polytopes in L, which cannot be extended to perfect (n+1)-dimensional centrally symmetric Delaunay polytopes. --Lattices, which have several orbits of non-isometric perfect Delaunay polytopes. Finally, we derived an upper bound for the covering radius of Λ_{24}(v)^{*}, which generalizes the Smith bound and we prove that it is met only by Λ_{23}^{*}, the best known lattice covering in R^{23}.

math.NT↗

arXiv2007

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.

math.NT↗

arXiv2007

Convexity of Hypersurfaces in Spherical Spaces

A spherical set is called convex if for every pair of its points there is at least one minimal geodesic segment that joins these points and lies in the set. We prove that for n >= 3 a complete locally-convex (topological) immersion of a connected (n-1)-manifold into the n-sphere is a surjection onto the boundary of a convex set.

math.MG↗

arXiv2007

An Efficient Local Approach to Convexity Testing of Piecewise-Linear Hypersurfaces

We show that a closed piecewise-linear hypersurface immersed in $R^n$ ($n\ge 3$) is the boundary of a convex body if and only if every point in the interior of each $(n-3)$-face has a neighborhood that lies on the boundary of some convex body; no assumptions about the hypersurface's topology are needed. We derive this criterion from our generalization of Van Heijenoort's (1952) theorem on locally convex hypersurfaces in $R^n$ to spherical spaces. We also give an easy-to-implement convexity testing algorithm, which is based on our criterion. For $R^3$ the number of arithmetic operations used by the algorithm is at most linear in the number of vertices, while in general it is at most linear in the number of incidences between the $(n-2)$-faces and $(n-3)$-faces. When the dimension $n$ is not fixed and only ring arithmetic is allowed, the algorithm still remains polynomial. Our method works in more general situations than the convexity verification algorithms developed by Mehlhorn et al. (1996) and Devillers et al. (1998) -- for example, our method does not require the input surface to be orientable, nor it requires the input data to include normal vectors to the facets that are oriented "in a coherent way". For $R^3$ the complexity of our algorithm is the same as that of previous algorithms; for higher dimensions there seems to be no clear winner, but our approach is the only one that easily handles inputs in which the facet normals are not known to be coherently oriented or are not given at all. Furthermore, our method can be extended to piecewise-polynomial surfaces of small degree.

cs.CG↗

arXiv2007

Perfect Delaunay Polytopes in Low Dimensions

A lattice Delaunay polytope is known as perfect if the only ellipsoid, that can be circumscribed about it, is its Delaunay sphere. Perfect Delaunay polytopes are in one-to-one correspondence with arithmetic equivalence classes of positive quadratic functions on the n-dimensional integral lattice that can be recovered, up to a scale factor, from the representations of its minimum. We develop a structural theory of such polytopes and describe all known perfect Delaunay polytopes in dimensions one through eight. We suspect that this list is complete.

math.MG↗

arXiv2006

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 polytope in each successive dimension starting at some initial dimension; we have been able to construct an infinite number of such infinite sequences. Perfect Delaunay polytopes play an important role in the theory of Delaunay polytopes, and in Voronoi's theory of lattice types.

math.NT↗

arXiv2005

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 sequences of perfect Delaunay polytopes, one perfect polytope in each successive dimension starting at some initial dimension; we have been able to construct an infinite number of such infinite sequences. Perfect Delaunay polytopes are intimately related to the theory of Delaunay polytopes, and to Voronoi's theory of lattice types.

math.NT↗

arXiv2005

Criteria for Balance in Abelian Gain Graphs, with Applications to Piecewise-Linear Geometry

A gain graph is a triple (G,h,H), where G is a connected graph with an arbitrary, but fixed, orientation of edges, H is a group, and h is a homomorphism from the free group on the edges of G to H. A gain graph is called balanced if the h-image of each closed walk on G is the identity. Consider a gain graph with abelian gain group having no odd torsion. If there is a basis of the graph's binary cycle space each of whose members can be lifted to a closed walk whose gain is the identity, then the gain graph is balanced, provided that the graph is finite or the group has no nontrivial infinitely 2-divisible elements. We apply this theorem to deduce a result on the projective geometry of piecewise-linear realizations of cell-decompositions of manifolds.

math.CO↗

arXiv2005

Supertopes

A perfect (Delaunay) ellipsoid is an ellipsoid in n-dimensional Euclidean space that does not contain integral points in its interior, but is uniquely defined by integral points that lie on its surface. A perfect Delaunay polytope with respect to a positive quadratic form f() is a polytope with integral vertices that is circumscribed by a perfect Delaunay ellipsoid with an equation whose quadratic part is f(). This document has been corrected on January 15, 2005. Note that it represents the state of the area as of the end of 2002. For recent research on perfect Delaunay polytopes see my recent preprint, with Erdahl and Ordine, math.NT/0408122 on ArXiv.org .

math.NT↗

arXiv2004

Cycle and Circle Tests of Balance in Gain Graphs: Forbidden Minors and Their Groups

We examine two criteria for balance of a gain graph, one based on binary cycles and one on circles. The graphs for which each criterion is valid depend on the set of allowed gain groups. The binary cycle test is invalid, except for forests, if any possible gain group has an element of odd order. Assuming all groups are allowed, or all abelian groups, or merely the cyclic group of order 3, we characterize, both constructively and by forbidden minors, the graphs for which the circle test is valid. It turns out that these three classes of groups have the same set of forbidden minors. The exact reason for the importance of the ternary cyclic group is not clear.

math.CO↗

arXiv2003

Fast Verification of Convexity of Piecewise-linear Surfaces

We show that a realization of a closed connected PL-manifold of dimension n-1 in n-dimensional Euclidean space (n>2) is the boundary of a convex polyhedron (finite or infinite) if and only if the interior of each (n-3)-face has a point, which has a neighborhood lying on the boundary of an n-dimensional convex body. No initial assumptions about the topology or orientability of the input surface are made. The theorem is derived from a refinement and generalization of Van Heijenoort's theorem on locally convex manifolds to spherical spaces. Our convexity criterion for PL-manifolds implies an easy polynomial-time algorithm for checking convexity of a given PL-surface in n-dimensional Euclidean or spherical space, n>2. The algorithm is worst case optimal with respect to both the number of operations and the algebraic degree. The algorithm works under significantly weaker assumptions and is easier to implement than convexity verification algorithms suggested by Mehlhorn et al (1996-1999), and Devillers et al.(1998). A paradigm of approximate convexity is suggested and a simplified algorithm of smaller degree and complexity is suggested for approximate floating point convexity verification.

cs.CG↗

arXiv2003

On locally convex PL-manifolds and fast verification of convexity

We show that a realization of a closed connected PL-manifold of dimension n-1 in Euclidean n-space (n>2) is the boundary of a convex polyhedron if and only if the interior of each (n-3)-face has a point, which has a neighborhood lying on the boundary of a convex n-dimensional body. This result is derived from a generalization of Van Heijenoort's theorem on locally convex manifolds to spherical spaces. We also give a brief analysis of how local convexity and topology of non-compact surfaces are related to global convexity in the hyperbolic space. Our convexity criterion for PL-manifolds imply an easy polynomial-time algorithm for checking convexity of a given closed compact PL-surface in Euclidean of spherical space of dimension n>2.

math.MG↗

arXiv2001

Voronoi-Dickson Hypothesis on Perfect Forms and L-types

George Voronoi (1908, 1909) introduced two important reduction methods for positive quadratic forms: the reduction with perfect forms, and the reduction with L-type domains, often called domains of Delaunay type. The first method is important in studies of dense lattice packings of spheres. The second method provides the key tools for finding the least dense lattice coverings with equal spheres in lower dimensions. In his investigations Voronoi heavily relied on that in dimensions less than 6 the partition of the cone of positive quadratic forms into L-types refines the partition of this cone into perfect domains. Voronoi conjectured implicitely and Dickson (1972) explicitely that the L-partition is always a refinement of the partition into perfect domains. This was proved for n =< 5 (Voronoi, Delaunay, Ryshkov, Baranovskii). We show that Voronoi-Dickson conjecture fails already in dimension 6.

math.NT↗

arXiv2001

An Infinite Series of Perfect Quadratic Forms and Big Delaunay Simplexes in Z^n

George Voronoi (1908-09) introduced two important reduction methods for positive quadratic forms: the reduction with perfect forms, and the reduction with L-type domains. A form is perfect if can be reconstructed from all representations of its arithmetic minimum. Two forms have the same L-type if Delaunay tilings of their lattices are affinely equivalent. Delaunay (1937-38) asked about possible relative volumes of lattice Delaunay simplexes. We construct an infinite series of Delaunay simplexes of relative volume n-3, the best known as of now. This series gives rise to a new infintie series of perfect forms TF_{n} with interesting properties, e.g. TF_{5}=D_{5}, TF_{6}=E*_{6}, TF_{7}=ϕ_{15}^{7}. For all n the domain of TF_{n} is adjacent to the domain of the 2-nd perfect form D_{n}. Perfect form TF_{n} is a direct n-dimensional generalization of Korkine and Zolotareff's 3-rd perfect form ϕ_{2}^{5} in 5 variables. It is likely that this form is equivalent to Anzin's (1991) form h_n.

math.MG↗