arXiv subjects

Robert Erdahl

Publications and source records attributed to Robert Erdahl.

7 recordsLinked to original sources

arXiv2015

Minkowski sum of a Voronoi parallelotope and a segment

By a {\em Voronoi parallelotope} $P(a)$ we mean a parallelotope determined by a non-negative quadratic form $a$. It was studied by Voronoi in his famous memoir. For a set of vectors $\mathcal P$, we call its {\em dual} a set of vectors ${\mathcal P}^*$ such that $\langle p,q\rangle\in\{0,\pm 1\}$ for all $p\in{\mathcal P}$ and $q\in{\mathcal P}^*$. We prove that Minkowski sum of a Voronoi parallelotope $P(a)$ and a segment is a Voronoi parallelotope $P(a+a_e)$ if and only if this segment is parallel to a vector $e$ of the dual of the set of normal vectors of all facets of $P(a)$, where $a_e(p)=b\langle e,p\rangle^2$ is a quadratic form of rank 1 related to the segment.

math.MG↗

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

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 .

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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↗