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An upper estimate for the lattice point enumerator

Published online by Cambridge University Press:  26 February 2010

P. Gritzmann  and
J. M. Wills
P. Gritzmann
Affiliation:
Math. Inst. Univ. Siegen, Hölderlinstr. 3, D-5900 Siegen, Fed. Rep. Germany.
J. M. Wills
Affiliation:
Math. Inst. Univ. Siegen, Hölderlinstr. 3, D-5900 Siegen, Fed. Rep. Germany.
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Extract

Since Minkowski [29] gave his famous lattice point theorem for centrally symmetric convex bodies, a theorem that turned out to be of fundamental importance in number theory, much effort has been made to obtain tight estimates for the number of lattice points of a given lattice in convex bodies in terms of the basic quermass-integrals Wo,…, Wd, whose eminent role shows in Hadwiger's functional theorem [14, 15, 16, see also 17, p. 221–225]. (For the discrete analogues of Wo,…, Wd see [2].) The present paper is concerned with an upper estimate of this kind.

Type
Research Article
Information
Mathematika , Volume 33 , Issue 2 , December 1986 , pp. 197 - 203
Copyright
Copyright © University College London 1986

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References

1.Betke, U. and Gritzmann, P.. An application of valuation theory to two problems in discrete geometry. Discrete Math., 58 (1986), 81–85.CrossRefGoogle Scholar
2.Betke, U. and Kneser, M.. Zerlegungen und Bewertungen von Gitterpolytopen. J. reine u. angew. Math., 358 (1985), 202–208.Google Scholar
3.Betke, U. and McMullen, P.. Lattice points in lattice polytopes. Monatsh. Math., 99 (1985), 253–265.CrossRefGoogle Scholar
4.Betke, U. and Wills, J. M.. Stetige und diskrete Funktionale konvexer Körper. In: Contributions to Geometry, ed. by Tölke, J. and Wills, J. M. (Birkhäuser, Basel 1979), 226–237.CrossRefGoogle Scholar
5.Blichfeldt, H. F.. Notes on geometry of numbers. Bull. Amer. Math. Soc., 27 (1921), 150–153.Google Scholar
6.Bokowski, J.. Gitterpunktanzahl und Parallelkörpervolumen von Eikörpern. Monatshefte Math., 79 (1975), 93–101.CrossRefGoogle Scholar
7.Bokowski, J., Hadwiger, H. and Wills, J. M.. Eine Ungleichung zwischen Volumen, Oberfläche und Gitterpunktanzahl konvexer Körper im n–dimensionalen euklidischen Raum. Math. Z., 127 (1972), 363–364.CrossRefGoogle Scholar
8.Coxeter, H. S. M.. Regular polytopes (Dover, New York, 1973).Google Scholar
9.Davenport, H.. On a principle of Lipschitz. J. London Math. Soc., 26 (1951), 179–183.CrossRefGoogle Scholar
10.Fejes, L.Töth. Research problem 13. Periodica Math. Hung., 6 (1975), 197–199.Google Scholar
11.Gritzmann, P.. Finite packing of equal balls. J London Math. Soc., (2), 33 (1986), 543–553.CrossRefGoogle Scholar
12.Gruber, P. M. and Lekkerkerker, C. G.. Geometry of Numbers, 2nd ed. (to appear); 1st ed. Lekkerkerker, C. G. (North-Holland, Amsterdam, 1969).Google Scholar
13.Grünbaum, B.. Convex Polytopes (Interscience, London, 1967).Google Scholar
14.Hadwiger, H.. Einige Anwendungen eines Funktionalsatzes fur konvexe Körper in der Räumlichen Integralgeometrie. Monatschefte Math., 54 (1950), 345–353.CrossRefGoogle Scholar
15.Hadwiger, H.. Beweis eines Funktionalsatzes für konvexe Körper. Abh. Math. Sem. Univ. Hamburg, 17 (1951), 69–76.CrossRefGoogle Scholar
16.Hadwiger, H.. Additive Funktionale k-dimensionaler Eichkörper I, II. Arch. Math., 3 (1952), 470–478; 4 (1953), 374–379.CrossRefGoogle Scholar
17.Hadwiger, H.. Vorlesungen über Inhalt, Oberfläche und hoperimetrie (Springer, Berlin, 1957).CrossRefGoogle Scholar
18.Hadwiger, H.. Gitterperiodische Punktmengen und Isoperimetrie. Monatshefte Math., 96 (1972), 410–418.CrossRefGoogle Scholar
19.Hadwiger, H.. Gitterpunktanzahl im Simplex und Wills'sche Vermutung. Math. Ann., 239 (1979), 271–288.CrossRefGoogle Scholar
20.Hammer, J.. Unsolved problems concerning lattice points (Pitman, London, 1977).Google Scholar
21.Höhne, R.. Zur Gitterpunktanzahl im Simplex. Math. Ann., 251 (1980), 269–276.CrossRefGoogle Scholar
22.Kabatjanski, G. A. and Levenstein, V. I.. Bounds for packings on the sphere and in space (Russian), Problemy Peredaci Informaticii, 14 (1978), 3–25; English transl. in Problems of information transmission, 14 (1978), 1–17.Google Scholar
23.Leech, J.. Notes on sphere packings. Can. J. Math., 19 (1967), 251–267.CrossRefGoogle Scholar
24.Macdonald, I. G.. The volume of a lattice polyhedron. Proc. Cambridge Phil. Soc, 59 (1963), 719–726.CrossRefGoogle Scholar
25.Macdonald, I. G.. Polynomials associated with finite cell complexes. J. London Math. Soc. (2), 4 (1971), 181–192.CrossRefGoogle Scholar
26.Mahler, K.. On lattice points in n-dimensional star bodies I. Existence theorems. Proc Royal Soc London, A, 187 (1946), 151–187.Google Scholar
27.McMullen, P.. Non-linear angle-sum relations for polyhedral cones and polytopes. Math. Proc. Camb. Phil. Soc., 78 (1975), 247–261.CrossRefGoogle Scholar
28.McMullen, P.. Valuations and Euler-type relations on certain classes of convex polytopes. Proc. London Math. Soc., (3), 35 (1977), 113–135.CrossRefGoogle Scholar
29.Minkowski, H.. Geometrie der Zahlen (Teubner, Leipzig, 1896); (Johnson Reprint Corp., New York, 1968).Google Scholar
30.Oler, N.. An inequality in the geometry of numbers. Ada Math., 105 (1961), 19–48.Google Scholar
31.Reeve, J. E.. On the volume of lattice polyhedra. Proc. London Math. Soc., (3), 7 (1957), 378–395.CrossRefGoogle Scholar
32.Reeve, J. E.. A further note on the volume of lattice polyhedra. J. London Math. Soc., 34 (1959), 57–62.CrossRefGoogle Scholar
33.Rogers, C. A.. The packing of equal spheres. Proc. London Math. Soc., (3), 8 (1958), 609–620.CrossRefGoogle Scholar
34.Rogers, C. A.. Packing and Covering (Cambridge University Press, 1964).Google Scholar
35.Schmidt, W. M.. Volume, surface area and the number of integer points covered by a convex set. Arch. Math., 23 (1972), 537–543.CrossRefGoogle Scholar
36.Wills, J. M.. Zur Gitterpunktanzahl konvexer Mengen. Elem. Math, 28 (1973), 57–63.Google Scholar
37.Zassenhaus, H.. Methoden und Probleme der modernen Algebra. (Selbstverlag, Universität Hamburg, 1947).Google Scholar
38.Zassenhaus, H.. Modern developments in the geometry of numbers. Bull. Atner. Math. Soc., 67 (1961), 427–439.CrossRefGoogle Scholar