Geodesic
On a problem in the geometry of numbers
Trudy Instituta matematiki, Tome 15 (2007) no. 1, pp. 111-117.

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In this paper, a problem of the geometry of numbers is considered. The notion of the defect is introduced, which helps to measure the difference between a basis in a lattice and a basis in its centering. Various estimates for the defect are obtained as well as some new problems are discussed. 
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A. M. Raigorodskii. On a problem in the geometry of numbers. Trudy Instituta matematiki, Tome 15 (2007) no. 1, pp. 111-117. http://geodesic.mathdoc.fr/item/TIMB_2007_15_1_a12/

[1] Cassels J.W.S., An introduction to the geometry of number, Springer-Verlag, Berlin et al., 1959 | Zbl

[2] Gruber P.M., Lekkerkerker C.G., Geometry of numbers, North-Holland, Amsterdam, 1987 | MR | Zbl

[3] Mordell L.J., “Lattice octahedra”, Canad. J. Math., 12 (1960), 297–302 | DOI | MR | Zbl

[4] Wessels U., Die Sätze von White und Mordell über kritische Gitter von Polytopen in die Dimensionen 4 und 5, Diplomarbeit, Mathematisches Institut der Ruhr-Universität Bochum, 1989

[5] Bantegnie R., “Le problème des octaèdres en dimension 5”, Acta Arithm., 14 (1968), 185–202 | MR | Zbl

[6] Ryshkov S.S., “On the problem of determination of the perfect quadratic forms in many variables”, Trudy Mat. Inst. Steklov, 142, 1976, 215–239 | MR | Zbl

[7] Zakharova N.V., “Centerings of 8-dimensional lattices that preserve a frame of successive minima”, Trudy Mat. Inst. Steklov, 152, 1980, 97–123 | MR | Zbl

[8] Moshchevitin N.G., “The defect of an admissible octahedron in a lattice”, Mat. Zametki, 58 (1995), 558–568 | MR | Zbl

[9] Raigorodskii A.M., “The defects of admissible balls and octahedra in a lattice, and systems of generic representatives”, Mat. Sbornik, 189:6 (1998), 117–141 | MR

[10] Minkowski H., Geometrie der Zahlen, Leipzig, Berlin, 1896

[11] Hlawka E., “Zur Geometrie der Zahlen”, Math. Z., 49 (1943), 285–312 | DOI | MR | Zbl

[12] Rogers C.A., “Lattice covering of space: The Minkowski–Hlawka theorem”, Proc. London Math. Soc., 3:8 (1958), 447–465 | DOI | MR | Zbl

[13] Schmidt W., “On the Minkowski–Hlawka theorem”, Illinois J. Math., 7 (1963), 18–23 | MR

[14] Kabatiansky G.A., Levenstein V.I., “Bounds for packings on a sphere and in space”, Problems in Information Transmission, 14 (1978), 1–17

[15] Raigorodskii A.M., “Systems of common representatives”, Fund. i Prikl. Mat., 5:3 (1999), 851–860 (in Russian) | MR

[16] Kuzyurin N.N., “Asymptotic investigation of the set covering problem”, Probl. Kibernet., 37, 1980, 19–56 | MR | Zbl

[17] Turán P., “Egy grafelmeletiszelsoertek feladatrol”, Mat. Fiz. Lapok, 1941, no. 48, 436–452 | MR

[18] Katona G., Nemetz T., Simonovitz M., “On a graph problem of Turan”, Mat. Lapok, 15 (1964), 228–238 (in Hungarian) | MR | Zbl

[19] Füredi Z., “Turán's type problems”, Surveys in Combinatorics, Proc. of the 13th British Combin. Conf., London Math. Soc. Lecture Note Series, 166, ed. A.D. Keedwell, Cambridge Univ. Press, 1991, 253–300 | MR

[20] Raigorodskii A.M., “A probabilistic approach to the problem of the defects of admissible sets in a lattice”, Mat. Zametki, 68:6 (2000), 910–916 | MR | Zbl

[21] Raigorodskii A.M., “The defect of admissible sets in a lattice, and systems of common representatives”, Grazer Math. Berichte, Beiträge zur zahlentheoretischen Analysis, 338, 1999, 31–62 | MR | Zbl