engleski

Lattice Packings and Coverings

A family of balls in Euclidean space is called a packing if for any two balls B and B' their interior do not self-intersect. It is called a covering if every point belong to at least one ball. We focus here on packings and coverings for which the calls are of the form x + B(0, R) with x belonging to a lattice L. If L is fixed then we adjust the value of R to a value R0 to find the best packing. Alternatively we can adjust the value of R to a value R1 to find the best covering. This allow us to define the packing density pack(L) and covering density cov(L) of L. The geometry of the function pack on the space of lattices has been elucidated by Minkovski, Voronoi and Ash. They showed that the function pack has no local minimum, that it is a Morse function and they give a characterization of the local maximum in terms of the algebraic notions of perfection and eutaxy. The covering function cov is much more complex. It has local minimum and local maximum and it is not a Morse function. We also characterize the local maximum of the covering density in terms of the corresponding notions of perfection and eutaxy this time for Delaunay polytope.

perfect form; Erdahl cone; eutaxy; design

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