Skip to main content
SpringerLink
Log in

On linear sections of lattice packings

  • Published:
Monatshefte für Mathematik Aims and scope Submit manuscript

Abstract

Ind-dimensional euclidean spaceE d letP be a lattice packing of subsets ofE d, and letH be ak-dimensional linear subspace ofE d (0<k<d). Then,P induces a packing inH consisting of all setsP∩H withP∈P. The relationship between the density of this packing inH and the density ofP is investigated. A result from the theory of uniform distribution of linear forms is used to prove an integral formula that enables one to evaluate the density of the induced packing inH (under suitable assumptions on the sets ofP and the functionals used to define the densities). It is shown that this result leads to explicit formulas for the averages of the induced densities under the rotation ofH. If the densities are taken with respect to the mean cross-sectional measures of convex bodies one obtains analogues of the integral geometric intersection formulas of Crofton.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Bezdek, A.: On the section of a lattice covering of balls. Rend. Circ. Mat. Palermo II. Ser. Suppl.3, 23–45 (1984).

    Google Scholar 

  2. Fejes-Tóth, G.: On the section of a packing of equal balls. Studia Sci. Math. Hungar.15, 487–489 (1980).

    Google Scholar 

  3. Groemer, H.: Existenzsätze für Lagerungen im Euklidischen Raum. Math. Zeitschr.81, 260–278 (1963).

    Google Scholar 

  4. Groemer, H.: Über ebene Schnitte von Lagerungen. Mh. Math.70, 213–222 (1966).

    Google Scholar 

  5. Groemer, H.: Über die mittlere Dichte von Massenverteilungen im Raum. Mh. Math.70, 437–443 (1966).

    Google Scholar 

  6. Groemer, H.: Eulersche Charakteristik, Projektionen und Quermassintegrale. Math. Ann.198, 23–56 (1972).

    Google Scholar 

  7. Hadwiger, H.: Vorlesungen über Inhalt, Oberfläche und Isoperimetrie. Berlin-Göttingen-Heidelberg: Springer. 1957.

    Google Scholar 

  8. Halmos, P. R.: Measure Theory. Princeton-New York-Toronto-London: Van Nostrand. 1950.

    Google Scholar 

  9. McMullen, P., Schneider, R.: Valuations on convex bodies. In: “Convexity and Its Applications”, pp. 170–247 (P. M. Gruber and J. M. Wills, editors). Basel-Boston-Stuttgart: Birkhäuser. 1983.

    Google Scholar 

  10. Santaló, L.: Integral Geometry and Geometric Probability Theory. Encyclopedia of Math. and Its Appl., Vol. 1. Reading, Mass.: Addison-Wesley. 1976.

    Google Scholar 

  11. Van der Corput, J. G.: Diophantische Ungleichungen I. Zur Gleichverteilung Modulo Eins. Acta Math.56, 373–456 (1931).

    Google Scholar 

  12. Weyl, H.: Über die Gleichverteilung von Zahlen mod. Eins. Math. Ann.77, 313–352 (1916) (=Ges. Abh. Bd. 1, pp. 563–599. Berlin-Heidelberg-New York: Springer. 1968.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, The University of Arizona, 85721, Tucson, Arizona, USA

    H. Groemer

Authors
  1. H. Groemer
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Dedicated to Professor E. Hlawka on the occasion of his seventieth birthday

Supported by National Science Foundation Research Grant DMS 8300825.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Groemer, H. On linear sections of lattice packings. Monatshefte für Mathematik 102, 199–216 (1986). https://doi.org/10.1007/BF01294599

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01294599

Keywords

Search

Navigation