Hostname: page-component-586b7cd67f-dlnhk Total loading time: 0 Render date: 2024-11-26T17:22:13.925Z Has data issue: false hasContentIssue false
  • English
  • Français

A new approach to covering

Part of: Discrete geometry

Published online by Cambridge University Press:  26 February 2010

Ulrich Betke ,
Martin Henk  and
Jörg M. Wills
Ulrich Betke
Affiliation:
Mathematisches Institut, Universität Siegen, Hölderlinstrasse 3, D-57068, Siegen, Germany.
Martin Henk
Affiliation:
Technische Universität Berlin, Sekr, MA6-1, Strasse des 17 Juni 136, D-10623 Berlin, Germany.
Jörg M. Wills
Affiliation:
Mathematisches Institut, Universität Siegen, Hölderlinstrasse 3, D-57068, Siegen, Germany.
Get access

Abstract

For finite coverings in euclidean d-space Ed we introduce a parametric density function. Here the parameter controls the influence of the boundary of the covered region to the density. This definition gives a new approach to covering which is similar to the approach for packing in [BHW1], [BHW2]. In this way we obtain a unified theory for finite and infinite covering and generalize similar results, which were developed by various authors since 1950 for d=2, to all dimensions.

MSC classification

Secondary: 52C17: Packing and covering in $n$ dimensions
Type
Research Article
Information
Mathematika , Volume 42 , Issue 2 , December 1995 , pp. 251 - 263
Copyright
Copyright © University College London 1995

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[BR].Batnbah, R. P. and Rogers, C. A.. Covering the plane with convex sets. J. London Math. Soc, 27 (1952), 304–314.Google Scholar
[BRZ].Bambah, R. P., Rogers, C. A. and Zassenhaus, H.. On coverings with convex domains. Ada Arithm., 9 (1964), 191–207.Google Scholar
[BW].Bambah, R. P. and Woods, A. C.. On plane coverings with convex domains. Mathematika, 18 (1971), 91–97.CrossRefGoogle Scholar
[BGW].Betke, U., Gritzmann, P. and Wills, J. M.. Slices of L. Fejes Toth's sausage conjecture. Mathematika, 29 (1982), 194–201.CrossRefGoogle Scholar
[BHW1].Betke, U., Henk, M. and Wills, J. M.. Finite and infinite packings. J. reine angew. Math., 453 (1994), 165–191.Google Scholar
[BHW2].Betke, U., Henk, M. and Wills, J. M.. Sausages are good packings. Discrete and Computational Geometry (to appear).Google Scholar
[BF].Bonnesen, T. and Fenchel, W.. Theorie der konvexen Körper (Springer, Berlin, 1934).Google Scholar
[FGW].Tóth, G. Fejes, Gritzmann, P. and Wills, J. M.. Sausage-skin problems for finite coverings. Mathematika, 31 (1984), 118–137.CrossRefGoogle Scholar
[FK].Tóth, G. Fejes and Kuperberg, W.. Packing and covering with convex sets, Ch. 3.3 In Handbook of Convex Geometry, edited by Gruber, P. M. and Wills, J. M. (North-Holland, Amsterdam, 1993).Google Scholar
[Gri].Gritzmann, P.. Ein Approximationssatz fur konvexe Körper. Geom. Dedicata, 19 (1985), 277–286.CrossRefGoogle Scholar
[GW1].Gritzmann, P. and Wills, J. M.. Finite packing and covering, Ch. 3.4 In Handbook of Convex Geometry, edited by Gruber, P. M. and Wills, J. M. (North-Holland, Amsterdam, 1993).Google Scholar
[GW2].Gritzmann, P. and Wills, J. M.. On two finite covering problems of Bambah, Rogers, Woods and Zassenhaus. Monatsh. Math., 99 (1985), 279–296.Google Scholar
[G].Gruber, P. M.. Aspects of approximation of convex bodies, Ch. 1.10 In Handbook of Convex Geometry, edited by Gruber, P. M. and Wills, J. M. (North-Holland, Amsterdam, 1993).Google Scholar
[GL].Gruber, P. M. and Lekkerkerker, C. G.. Geometry of Numbers (North-Holland, Amsterdam, 1987).Google Scholar
[Gr].Groemer, H.. Über die Einlagerungen von Kreisen in einen konvexen Bereich. Math. Z., 73 (1960), 285–294.CrossRefGoogle Scholar
[H].Henk, M.. Finite and Infinite Packings. (Habilitationsschrift, Siegen, 1994).Google Scholar
[O].Oler, N.. An inequality in the geometry of numbers. Ada Math., 105 (1961), 19–48.Google Scholar
[R].Rogers, C. A.. Packing and Covering. (Camb. Univ. Press, Cambridge, 1964).Google Scholar
[S].Schneider, R.. Convex Bodies: The Brunn-Minkowski Theory (Camb. Univ. Press, Cambridge, 1993).CrossRefGoogle Scholar