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- Robert M. Erdahl
- Discrete & Computational Geometry
- 1992

- Robert M. Erdahl
- Eur. J. Comb.
- 1999

- Robert M. Erdahl, Sergei S. Ryshkov
- Eur. J. Comb.
- 1994

Georges 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 it 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… (More)

A polytope D, whose vertices belong to a lattice of rank d, is Delaunay if it can be circumscribed by an ellipsoid E with interior free of lattice points, and so that the vertices of D are the only lattice points on the quadratic surface E. If in addition E is uniquely determined by D, we call D a perfect Delaunay polytope. Thus, in the perfect case, the… (More)

- Mathieu Dutour, Robert M. Erdahl, Konstantin A. Rybnikov
- 2007

A lattice Delaunay polytope is perfect if its Delaunay sphere is its only circumscribed ellipsoid. A perfect Delaunay polytope naturally corresponds to a positive quadratic function on Z that can be recovered uniquely from the data consisting of its minimum and all points of Z where this minimum is achieved – a quadratic function with this uniqueness… (More)

- Robert M. Erdahl, Konstantin A. Rybnikov, Sergei S. Ryshkov
- Eur. J. Comb.
- 2001

We show how a d-stress on a piecewise-linear realization of an oriented (non-simplicial, in general) d-manifold in Rd naturally induces stresses of lower dimensions on this manifold, and discuss implications of this construction to the analysis of self-stresses in spatial frameworks. The mappings we construct are not linear, but polynomial. In the 1860–70s… (More)

George Voronoi (1908, 1909) introduced two important reduction methods for positive quadratic forms: the reduction with perfect forms, and the reduction with Ltype 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… (More)

- Robert M. Erdahl
- ISVD
- 2006

I will consider two outstanding problems in the theory of Delaunay and Voronoi tilings for lattices. There is a new classification problem that has arisen from a new structure theorem. Sergei Ryshkov proved that: The Minkowski sum of two Voronoi polytopes is a Voronoi polytope, if and only if the corresponding Delaunay tilings are commensurate. This result… (More)

In his last two papers Georges Voronoi introduced two tilings for the cone of metrical forms for lattices, the tiling by perfect domains and the tiling by lattice type domains. Both are facet-to-facet tilings by polyhedral subcones, and invariant with respect to the natural action of GL(n, Z) on Sym(n, R). In working out the details of his theory of lattice… (More)