Article contents
A new approach to covering
Part of:
Discrete geometry
Published online by Cambridge University Press: 26 February 2010
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
- Type
- Research Article
- Information
- Copyright
- Copyright © University College London 1995
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
[S].Schneider, R.. Convex Bodies: The Brunn-Minkowski Theory (Camb. Univ. Press, Cambridge, 1993).CrossRefGoogle Scholar
- 5
- Cited by