Naslov
Lattice packings and coverings

Autori
Dutour Sikirić, Mathieu ; Vallentin, Frank ; Schuermann, Achill

Vrsta, podvrsta i kategorija rada
Sažeci sa skupova, pp prezentacija, znanstveni

Izvornik
ISM Symposium, Stochastic models and discrete geometry, The institute of statistical mathematics / - , 2010

Skup
ISM Symposium, Stochastic models and discrete geometry, The institute of statistical mathematics

Mjesto i datum
Tokyo, Japan, 01.03.2010. - 02.03.2010

Vrsta sudjelovanja
Plenarno

Vrsta recenzije
Međunarodna recenzija

Ključne riječi
Delaunay polytopes; covering; packing; extremal problem

Sažetak
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 allows 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 Minkowski, 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. We prove that it is not a Morse function. The natural optimization problem for covering is to find local minima of cov but we show that the local covering maxima of cov are also interesting. In particular we prove the following theorem: A lattice $L$ is a covering maxima if and only if all of its Delaunay polytopes of maximal circumradius are perfect and eutactic. Examples of perfect and eutactic Delaunay polytopes are 2(21) and 3(21) in the root lattices E6 and E7, which are thus local covering maxima.

Izvorni jezik
Engleski

Znanstvena područja
Matematika

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