Hostname: page-component-848d4c4894-x5gtn Total loading time: 0 Render date: 2024-05-19T00:51:42.089Z Has data issue: false hasContentIssue false
  • English
  • Français

The realization of distances within sets in Euclidean space

Part of: Designs and configurations

Published online by Cambridge University Press:  26 February 2010

D. G. Larman  and
C. A. Rogers
D. G. Larman
Affiliation:
Department of Mathematics, University College London.
C. A. Rogers
Affiliation:
Department of Mathematics, University College London.
Get access

Extract

In 1944 and 1945 H. Hadwiger [1, 2] proved the following theorems.

Theorem A. Let En be covered by n + 1 closed sets. Then there is one of the sets, within which all distances are realized.

Theorem B. Let En be covered by 4n−3 closed sets that are all mutually congruent. Then all distances are realized within each set.

Here a distance d is realized within a set S, if there are points x, y in S at distance d apart.

MSC classification

Secondary: 05B99: None of the above, but in this section
Type
Research Article
Information
Mathematika , Volume 19 , Issue 1 , June 1972 , pp. 1 - 24
Copyright
Copyright © University College London 1972

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

1.Hadwiger, H., “Ein Überdeckungssätze für den Euklidischen Raum”, Portugaliae Math., 4(1944), 140–144.Google Scholar
2.Hadwiger, H., “Ueberdeckung des Euklidischen Raumes durch Kongruente Mengen”, Portugaliae Math., 4 (1945), 238–249.Google Scholar
3.Raiskii, D. E., The realisation of all distances in a decomposition of the space R R into n+1 parts (Russian). Mat. Zametki, 7 (1970), 319–323.Google Scholar
4.Raiskii, D. E., “The realisation of all distances in a decomposition of the space Rn into n+1 parts.” Math. Notes, 7 (1970), 194–196.CrossRefGoogle Scholar
5.Woodall, D. R., “Distances realized by sets covering the plane.” J. Combinatorial Theory, A, to appear.Google Scholar
6.Moser, L. and Moser, W., “Problem 10”, Canadian Math. Bull., 4 (1961), 187–189.Google Scholar
7.Gosset, T., “On the regular and semi-regular figures in space of n dimensions.” Messenger of Mathematics, 29 (1900), 43–48.Google Scholar
8.Coxeter, H. S. M., “The pure Archimedean polytopes in six and seven dimensions”, Proc. Camb. Phil. Soc, 24 (1927), 1–9.CrossRefGoogle Scholar
9.Rankin, R. A., “The closest packing of spherical caps in n dimensions.” Proc. Glasgow Math. Assoc., 2 (1955), 139–144.CrossRefGoogle Scholar
10.Sarkadi, K. and Szele, T., “Feladat 20”, Mat. Lapok, 2 (1951), 76–77.Google Scholar
11.Aczél, J. and Szele, T., “Feladat 35”, Mat. Lapok, 3 (1952), 94–95.Google Scholar
12.Leech, J., “Some sphere packings in higher space.” Canadian J. Math., 16 (1964), 657–682.CrossRefGoogle Scholar
13.Leech, J., “Notes on sphere packings.” Canadian J. Math., 19 (1967), 251–267.CrossRefGoogle Scholar
14.Conway, J. H., “A group of order 8, 315, 553, 613, 086, 720, 000.” Bull. London Math. Soc, 1 (1969), 79–88.CrossRefGoogle Scholar
15.Hilton, A. J. W. and Milner, E. C., “Intersection theorems for systems of finite sets.” Quart, J. Math., Oxford (2), 18 (1967), 369–384.CrossRefGoogle Scholar
16.Kleitman, D. J., “On a conjecture of Milner on k graphs with non-disjoint edges.” J. Combinatorial Theory, 5 (1968), 153–156.CrossRefGoogle Scholar
17.Butler, G. J., “Simultaneous packing and covering in Euclidean space.” Proc. London Math. Soc., to appear.Google Scholar
18.Erdós, P. and Rogers, C. A., “Covering space with convex bodies.” Ada Arithmetica, 7 (1962), 281–285.CrossRefGoogle Scholar