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A counterpart to Oler's lattice-point theorem

Published online by Cambridge University Press:  26 February 2010

J. M. Wills
J. M. Wills
Affiliation:
Math. Inst. Univ. Siegen, Hoelderlinstr. 3, D-5900 Siegen, Fed. Rep. Germany.
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Abstract

Oler's lattice-point theorem gives a sharp upper bound for the lattice-point enumerator GΛ of a certain class of lattices in the plane. We give a sharp lower bound for GΛ of the corresponding class of lattices in all dimensions. This result is closely related to the Blichfeldt-van der Corputgeneralization of Minkowski's fundamental lattice-point theorem.

Type
Research Article
Information
Mathematika , Volume 36 , Issue 2 , December 1989 , pp. 216 - 220
Copyright
Copyright © University College London 1989

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References

BHW.Bokowski, J., Hadwiger, H. and Wills, J. M.. Eine Ungleichung zwischen Volumen, Oberfläche und Gitterpunktanzahl konvexer Kőrper im n-dimensionalen euklidischen Raum. Math. Z, 127 (1972), 363–364.CrossRefGoogle Scholar
CS.Conway, J. H. and Sloane, N. J. A.. Sphere packings, lattices and groups (Springer, New York, 1988).CrossRefGoogle Scholar
FG.Folkman, J. H. and Graham, R. L.. A packing inequality for compact convex subsets of the plane. Canad. Math. Bull., 12 (1969), 745–752.CrossRefGoogle Scholar
G.Gritzmann, P.. Finite packing of equal balls. j London Math. Soc, 33 (1986), 543–553.CrossRefGoogle Scholar
GW.Gritzmann, P. and Wills, J. M.. An upper estimate for the lattice-point enumerator. Mathematika, 33 (1986), 197–203.CrossRefGoogle Scholar
GL.Gruber, P. M. and Lekkerkerker, C. G.. Geometry of numbers (North-Holland, Amsterdam, 1987).Google Scholar
H.Hadwiger, H.. Vorlesungen über Inhalt, Oberfläche und Isoperimetrie (Springer, Berlin, 1975).Google Scholar
M.McMullen, P.. Non-linear angle-sum relations for polyhedral cones and polytopes. Math. Proc. Camb. Phil. Soc, 78 (1975), 247–261.CrossRefGoogle Scholar
O.Oler, N.. An inequality in the geometry of numbers. Acta Math., 105 (1961), 19–48.CrossRefGoogle Scholar
R.Rogers, C. A.. Packing and covering (Cambridge Univ. Press, 1964).Google Scholar
W.Wills, J. M.. Kugellagerungen und Konvexgeometrie. To appear in Jahresbericht Deutsche Math. Ver., 92 (1990).Google Scholar
Z.Zassenhaus, H.. Modern development in the geometry of numbers. Bull. Amer. Math. Soc, 67 (1961), 427–439.CrossRefGoogle Scholar