# Inhomogeneous extreme forms

Mathieu Dutour Sikirić^{[1]}; Achill Schürmann^{[2]}; Frank Vallentin^{[3]}

- [1] Rudjer Bosković Institute Bijenicka 54, 10000 Zagreb (Croatia)
- [2] Universität Rostock Institut für Mathematik 18051 Rostock (Germany)
- [3] Technical University of Delft Delft Institute of Applied Mathematics P.O. Box 5031, 2600 GA Delft (The Netherlands)

Annales de l’institut Fourier (2012)

- Volume: 62, Issue: 6, page 2227-2255
- ISSN: 0373-0956

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topDutour Sikirić, Mathieu, Schürmann, Achill, and Vallentin, Frank. "Inhomogeneous extreme forms." Annales de l’institut Fourier 62.6 (2012): 2227-2255. <http://eudml.org/doc/251090>.

@article{DutourSikirić2012,

abstract = {G.F. Voronoi (1868–1908) wrote two memoirs in which he describes two reduction theories for lattices, well-suited for sphere packing and covering problems. In his first memoir a characterization of locally most economic packings is given, but a corresponding result for coverings has been missing. In this paper we bridge the two classical memoirs.By looking at the covering problem from a different perspective, we discover the missing analogue. Instead of trying to find lattices giving economical coverings we consider lattices giving, at least locally, very uneconomical ones. We classify local covering maxima up to dimension $6$ and prove their existence in all dimensions beyond.New phenomena arise: Many highly symmetric lattices turn out to give uneconomical coverings; the covering density function is not a topological Morse function. Both phenomena are in sharp contrast with the packing problem.},

affiliation = {Rudjer Bosković Institute Bijenicka 54, 10000 Zagreb (Croatia); Universität Rostock Institut für Mathematik 18051 Rostock (Germany); Technical University of Delft Delft Institute of Applied Mathematics P.O. Box 5031, 2600 GA Delft (The Netherlands)},

author = {Dutour Sikirić, Mathieu, Schürmann, Achill, Vallentin, Frank},

journal = {Annales de l’institut Fourier},

keywords = {lattices; Delone polytopes; spherical $t$-designs; sphere packing; sphere covering; Voronoi reduction theory; spherical -designs},

language = {eng},

number = {6},

pages = {2227-2255},

publisher = {Association des Annales de l’institut Fourier},

title = {Inhomogeneous extreme forms},

url = {http://eudml.org/doc/251090},

volume = {62},

year = {2012},

}

TY - JOUR

AU - Dutour Sikirić, Mathieu

AU - Schürmann, Achill

AU - Vallentin, Frank

TI - Inhomogeneous extreme forms

JO - Annales de l’institut Fourier

PY - 2012

PB - Association des Annales de l’institut Fourier

VL - 62

IS - 6

SP - 2227

EP - 2255

AB - G.F. Voronoi (1868–1908) wrote two memoirs in which he describes two reduction theories for lattices, well-suited for sphere packing and covering problems. In his first memoir a characterization of locally most economic packings is given, but a corresponding result for coverings has been missing. In this paper we bridge the two classical memoirs.By looking at the covering problem from a different perspective, we discover the missing analogue. Instead of trying to find lattices giving economical coverings we consider lattices giving, at least locally, very uneconomical ones. We classify local covering maxima up to dimension $6$ and prove their existence in all dimensions beyond.New phenomena arise: Many highly symmetric lattices turn out to give uneconomical coverings; the covering density function is not a topological Morse function. Both phenomena are in sharp contrast with the packing problem.

LA - eng

KW - lattices; Delone polytopes; spherical $t$-designs; sphere packing; sphere covering; Voronoi reduction theory; spherical -designs

UR - http://eudml.org/doc/251090

ER -

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